Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations

Abstract

This paper develops the mathematical apparatus of studying control problems for the stationary model of magnetic hydrodynamics of viscous heat-conducting fluid in the Boussinesq approximation. These problems are formulated as problems of conditional minimization of special cost functionals by weak solutions of the original boundary value problem. The model under consideration consists of the Navier–Stokes equations, the Maxwell equations without displacement currents, the generalized Ohm’s law for a moving medium and the convection-diffusion equation for temperature. These relations are nonlinearly connected via the Lorentz force, buoyancy force in the Boussinesq approximation and convective heat transfer. Results concerning the existence and uniqueness of the solution of the original boundary value problem and of its generalized linear analog are presented. The global solvability of the control problem under study is proved and the optimality system is derived. Sufficient conditions on the data are established which ensure local uniqueness and stability of solutions of the control problems under study with respect to small perturbations of the cost functional to be minimized and one of the given functions. We stress that the unique stability estimates obtained in the paper have a clear mathematical structure and intrinsic beauty.

1. Introduction and Formulation of the Boundary Value Problem

In recent years, the theory of control of hydrodynamic, thermal and electromagnetic fields in liquid media has been intensively developed. One of the goals of the theory is to establish the most effective mechanisms for controlling physical fields in continuous media. Mathematical modeling of this type of problem includes three components: a goal, control parameters used to achieve the desired goal, and constraints that must be satisfied by the state of the system and controls under consideration. The role of constraints is usually played by the equations of the continuum model under consideration: hydrodynamics, magnetic hydrodynamics (MHD), heat and mass transfer, electromagnetism, etc., together with boundary and initial conditions, while the desired goal is achieved by minimizing certain cost functionals.

In hydrodynamics, the problem of reducing drag forces in a viscous fluid has always been a topical problem (see Gad-el-Hak [1]). In thermal convection, the problems of controlling the flow regime of a viscous heat-conducting fluid by means of various control actions, for example, heat sources, as well as the problems of minimizing temperature gradients or maximum temperatures in certain parts of the flow region are of interest (see Alifanov [2]). In engineering ecology, control problems arose when solving the urgent problem of environmental protection from anthropogenic impact (see Marchuk [3]).

In magnetic hydrodynamics, control problems have an important applied significance. Historically, the problems of controlling MHD flows arose first in metallurgy and foundry production when developing optimal technologies for non-contact electromagnetic stirring of molten metals [4,5], as well as in the nuclear industry when creating efficient liquid-metal cooling systems for nuclear power units [6,7]. Then, to the need to solve control problems led the problems arising in the creation of installations for industrial crystal growth by melting and dissolution methods [8,9] and the development of new submarine engines (see Convert [10]).

When solving the mentioned problems of magnetic hydrodynamics, the main attention was paid to the applied aspects of the developed theory. To a lesser extent, theoretical issues concerning the proof of solvability, existence and stability of solutions to the control problems for MHD models under consideration have been investigated. The present paper fills this gap. Its goal is to develop a mathematical apparatus for solving control problems for a stationary magnetohydrodynamics-Boussinesq model considered in the bounded domain $\mathsf{\Omega }$ of the space ${\mathbb{R}}^3$ with boundary $\mathsf{\Gamma }=\partial \mathsf{\Omega }$ consisting of two parts ${\mathsf{\Gamma }}_D$ and ${\mathsf{\Gamma }}_N$. The mentioned system is described by the following relations:

$$-\nu \mathsf{\Delta }\mathbf{u}+(\mathbf{u}·\nabla )\mathbf{u}+\nabla p-\kappa \mathrm{rot}\mathbf{H}\times \mathbf{H}=\mathbf{f}-\mathbf{b}T,\mathrm{div}\mathbf{u}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\mathbf{b}=b\mathbf{G},$$

(1)

$${\nu }_1\mathrm{rot}\mathbf{H}-\mathbf{E}+\kappa \mathbf{H}\times \mathbf{u}={\nu }_1\mathbf{j},\mathrm{div}\mathbf{H}=0,\mathrm{rot}\mathbf{E}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\kappa =\mu /{\rho }_0,{\nu }_1=\kappa {\nu }_m,$$

(2)

$$-\lambda \mathsf{\Delta }T+\mathbf{u}·\nabla T=f\mathrm{i}\mathrm{n}\mathsf{\Omega },$$

(3)

$${\mathbf{u}|}_{\mathsf{\Gamma }}=\mathbf{g},\mathbf{H}·\mathbf{n}{{|}_{\mathsf{\Gamma }}=q,\mathbf{E}\times \mathbf{n}|}_{\mathsf{\Gamma }}=\mathbf{k},$$

(4)

$$T=\psi \mathrm{o}\mathrm{n}{\mathsf{\Gamma }}_D,\lambda (\partial T/\partial n+\alpha T)=\chi \mathrm{o}\mathrm{n}{\mathsf{\Gamma }}_N,$$

(5)

describing the motion of a viscous heat-conducting and electrically conducting fluid.

The usual designations for this model are used here (see the textbook by Shercliff [11]): $\mathbf{u}$ and $\mathbf{H}$ are vectors of velocity and magnetic field strength, $\mathbf{E}={\mathbf{E}}^{\prime }/{\rho }_0$, $p=P/{\rho }_0$, where ${\mathbf{E}}^{\prime }$ is the vector of electric field strength, P is pressure, ${\rho }_0=\mathrm{const}$ is fluid density, T is temperature, $\nu$, ${\nu }_m$, $\sigma$ and $\lambda$ are constant coefficients of kinematic viscosity, magnetic viscosity, electrical conductivity and thermal diffusivity, $\mu ={\mu }^0\tilde{\mu }$, where ${\mu }^0$ is a vacuum magnetic permeability, $\tilde{\mu }=\mathrm{const}$ is a relative magnetic permeability, $\mathbf{G}$ is the free-fall acceleration vector, $\mathbf{f}$ is the bulk density of external forces, f is the volumetric density of heat sources, $\mathbf{j}$ is the vector of volume density of outward currents, b is the coefficient of thermal expansion, $\mathbf{g},\mathbf{k},q,\psi$ and $\alpha$, $\chi$ are functions defined on $\mathsf{\Gamma }$, ${\mathsf{\Gamma }}_D$, ${\mathsf{\Gamma }}_N$. The summand $\mathbf{b}T=bT\mathbf{G}$ makes sense of the buoyancy force in the Boussinesq approximation. With this in mind, below we will refer to the model (1)–(3) as the MHD-Boussinesq model or the MHDB model for brevity, and to the problem (1)–(5) itself as Problem 1. All quantities included in (1)–(5) are considered dimensional with all equations written in the SI unit system.

Theoretical analysis of control problems for MHDB models of a heat-conducting fluid is associated with a number of difficulties. The main difficulty is caused by the nonlinearity of the Navier–Stokes equations included in any MHDB model. As a consequence, the control problems are in general non-convex and multi-extremal. Another difficulty comes from the “multidisciplinarity” of MHDB models, which include, in addition to hydrodynamic processes, convective and diffusive heat transport processes as well as electromagnetic phenomena. Even though the behavior of each of the above processes separately is studied in detail, the study of their total interaction presents significant difficulties due to the nonlinearity of MHDB models.

In case $b=0$, the problem (1)–(5) is split into two problems: one of them, described by relations (1), (2), (4) at $\mathbf{b}=0$, is a boundary value problem for the stationary MHD model of a viscous non-thermal fluid. Its solvability is proved in Alekseev [12] in the case of $\mathbf{g}=0$, and in Alekseev [13] in the case when the vector $\mathbf{g}$ in (4) is tangential (see also the book by Alekseev ([14], Chapter 6), where, in addition, the corresponding control problems are studied).

In Villamizar-Roa [15] very weak solutions of MHD equations are investigated. The papers by Gunzburger et al. and Schotzau, respectively, refs. [16,17] are devoted to the development of numerical algorithms for solving boundary value problems for stationary MHD equations. The articles by Gunzburger et al. and Ravindran, respectively, refs. [18,19,20] study control problems for MHD models. In Refs. [21,22,23,24,25,26,27,28] control problems for stationary or nonstationary MHD models are investigated.

A number of papers are devoted to the study of solvability of boundary value (and control) problems for MHD Equations (1) and (2) at $b=0$ considered under boundary conditions different from those in (4). Let us mention among them the papers [29,30,31] considered under the so-called non-standard boundary conditions for velocity or pressure, as well as the papers by Meir and Alekseev et al., respectively, refs. [32,33,34] considered under mixed boundary conditions for the magnetic field. The papers [35,36,37,38,39,40,41,42,43] are devoted to the study of the solvability of boundary value and control problems for stationary or nonstationary MHD-Boussinesq models. The well-posedness of free boundary problems in non-relativistic and relativistic ideal compressible magnetohydrodynamics is studied in [44,45]. The papers [46,47,48,49,50,51,52,53,54,55,56,57] are devoted to the study of the solvability of control problems for close heat and mass transfer models.

We emphasize that control problems for MHDB flows of electrically and thermally conducting liquids have played an important applied role in the development of optimal technologies for the electromagnetic stirring of liquid metals in metallurgy [4,5], in the development of efficient cooling mechanisms for nuclear reactors [6,7], in the development of efficient crystal growth methods [8,9] and in the design of submarine engines (see Convert [10]).

In papers by Alekseev [34,42] an original approach to the study of boundary value problems for the MHD equations of viscous incompressible fluid was proposed. The use of this approach allowed us to obtain results on the solvability of boundary value problems under weakened conditions on the smoothness of boundary data for the magnetic field. The same idea proved useful when studying control problems for the MHD model (see Alekseev [58]). Its application allowed us to develop in Alekseev [58] an effective mathematical apparatus for the study of multiparameter control problems for the MHD model with minimal requirements for the smoothness of the boundary controls. Their role is played by the normal and tangential components of the magnetic field determined on different parts of the boundary of the flow region. The use of this theory allowed us to prove the uniqueness and stability of optimal solutions in the practically important case when boundary controls can be chosen from classes of integrable functions with squares instead of the commonly used trace spaces.

We also mention papers [59,60,61,62,63,64,65,66,67,68] which touch upon issues close to the subject of our paper from fields of electromagnetism, acoustics, nonlinear diffusion, engineering mechanics, viscoelasticity.

The present work continues to develop the direction started in the previous papers by Alekseev [34,42,58]. Its goal is to develop the mathematical apparatus for the study of control problems for the MHD-Boussinesq model of a viscous heat-conducting fluid and to apply the developed apparatus to prove the solvability of the control problems under consideration and to derive unique stability estimates of optimal solutions for certain control problems.

In view of the above, the present paper is organized as follows. Section 2 defines the function spaces and provides some additional facts that will be further used when developing the optimal control theory for the MHDB model under study. Section 3 presents Lemma 4 about the existence of liftings of all three main components (velocity, magnetic field, and temperature) of the solution of the boundary value problem under consideration. In addition, results on the existence and uniqueness of the solution of the boundary value problem and of its generalized linear analog are given. In Section 4, the main optimal control problem is formulated, its solvability is proved, an optimality system describing the necessary first-order optimality conditions is derived, and its properties are discussed. In Section 5 additional properties of optimal solutions are established based on the analysis of the optimality system. In Section 6, we prove the local uniqueness and stability of solutions to certain control problems for hydrodynamic and temperature cost functionals. Finally, Section 7 and Section 8 contain a discussion and brief summary of the results obtained in the paper.

2. Functional Spaces and Supporting Information

As usual, when studying the solvability of boundary value and control problems for models of MHD, we will use the function spaces $H^s\left(D\right)$, $s\in \mathbb{R}$, and $L^r\left(D\right)$, $1\le r<\infty$, where D represents the domain $\mathsf{\Omega }$ or its subset Q, or the boundary $\mathsf{\Gamma }$ or some part ${\mathsf{\Gamma }}_0$ of $\mathsf{\Gamma }$. Respective spaces of vector-functions $\mathbf{u}=(u_1,u_2,u_3)$ will be denoted by ${\mathbf{H}}^s\left(D\right)\equiv H^s{\left(D\right)}^3$. By ${(·,·)}_{s,D}$, ${\parallel ·\parallel }_{s,D}$ or ${|·|}_{s,D}$ the scalar product, norm or semi-norm in $H^s\left(D\right)$ or ${\mathbf{H}}^s\left(D\right)$ will be denoted, respectively. Scalar products and norms in $L^2\left(Q\right)$ or in $L^2\left(\mathsf{\Omega }\right)$ are denoted by ${(·,·)}_Q$, ${\parallel ·\parallel }_Q$ or by $(·,·)$ and ${\parallel ·\parallel }_{\mathsf{\Omega }}$, respectively. We make the following assumptions on $\mathsf{\Omega }$ and on the partition $\{{\mathsf{\Gamma }}_D,{\mathsf{\Gamma }}_N\}$ of the boundary $\mathsf{\Gamma }$:

Hypothesis 1.

Ω is a bounded domain in ${\mathbb{R}}^3$ with boundary $\mathsf{\Gamma }\in C^{1,1}$, and the open parts ${\mathsf{\Gamma }}_D$ and ${\mathsf{\Gamma }}_N$ of the boundary Γ satisfy the conditions: $meas{\mathsf{\Gamma }}_D>0$, ${\mathsf{\Gamma }}_D\cap {\mathsf{\Gamma }}_N=\varnothing$, $\mathsf{\Gamma }={\overline{\mathsf{\Gamma }}}_D\cup {\overline{\mathsf{\Gamma }}}_N$.

Hypothesis 2.

Ω is a domain in ${\mathbb{R}}^3$ with boundary Γ consisting of $p_0+1$ connected components ${\mathsf{\Gamma }}^0,{\mathsf{\Gamma }}^1,\dots ,{\mathsf{\Gamma }}^{p_0}$, and there exist non-intersecting manifolds ${\mathsf{\Sigma }}_i\in C^2$, $i=1,\dots ,q_0$ such that the domain $\tilde{\mathsf{\Omega }}=\mathsf{\Omega }\setminus {\displaystyle \bigcup _{i=1}^{q_0}}{\mathsf{\Sigma }}_i$ is simply connected and Lipschitzian.

The numbers $q_0$ and $p_0$ included in Hypothesis 2 are called by the first and second Betti numbers, respectively. Thus $p_0=0$ if and only if the boundary $\mathsf{\Gamma }$ is connected while $q_0=0$ if and only if $\mathsf{\Omega }$ is a simply connected domain. Typical examples of domains $\mathsf{\Omega }$ are shown in Figure 1. Figure 1a shows a simply connected domain $\mathsf{\Omega }$ with connected boundary $\mathsf{\Gamma }={\mathsf{\Gamma }}_D\cup {\mathsf{\Gamma }}_N$ (it corresponds to the case $p_0=0$ and $q_0=0$). Figure 1b shows a simply connected domain $\mathsf{\Omega }$ with non-connected boundary $\mathsf{\Gamma }={\mathsf{\Gamma }}_D\cup {\mathsf{\Gamma }}_N$ ($p_0=2$ and $q_0=0$). Figure 1c shows a multi-connected toroidal domain $\mathsf{\Omega }$ with connected boundary $\mathsf{\Gamma }={\mathsf{\Gamma }}_D\cup {\mathsf{\Gamma }}_N$ ($p_0=0$ and $q_0=1$). In general, $q_0$ indicates the number of surfaces (or cuts) ${\mathsf{\Sigma }}_i$ that need to be carried out in the domain $\mathsf{\Omega }$ in order to make it simply connected. Thus, in Figure 1c, the cross-section of the torus is chosen as the mentioned section ${\mathsf{\Sigma }}_1$. Figure 1d shows a domain $\mathsf{\Omega }$ having the form of a hollow cylinder with connected boundary $\mathsf{\Gamma }={\mathsf{\Gamma }}_D\cup {\mathsf{\Gamma }}_N$ ($p_0=0$ and $q_0=1$). From a physical point of view, Figure 1d demonstrates a schematic (simplified) illustration of a component part of a coaxial induction MHD generator with boundaries ${\mathsf{\Gamma }}_D$ and ${\mathsf{\Gamma }}_N$ (see the book by Vatazhin et al. [69]).

Let us consider the following subspaces of the space ${\mathbf{L}}^2\left(\mathsf{\Omega }\right)$:

$$H(\mathrm{rot},\mathsf{\Omega })=\{\mathbf{h}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right):\mathrm{rot}\mathbf{h}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)\},H_{DC}=\{\mathbf{h}\in H(\mathrm{rot},\mathsf{\Omega }):\mathrm{div}\mathbf{h}\in L^2\left(\mathsf{\Omega }\right)\}.$$

The latter space is Hilbertian with the Hilbert norm

$${\parallel \mathbf{v}\parallel }_{DC}^2=l^{-2}{\parallel \mathbf{v}\parallel }_{\mathsf{\Omega }}^2+{\parallel \mathrm{rot}\mathbf{v}\parallel }_{\mathsf{\Omega }}^2+{\parallel \mathrm{div}\mathbf{v}\parallel }_{\mathsf{\Omega }}^2.$$

(6)

Here, l is a dimensional coefficient having dimension $\left[l\right]=L_0$ and the value is equal to 1, $L_0$ denotes the SI dimension of length defined in meters (see below for details on the dimensions of the main parameters).

Let us define two subspaces of harmonic vectors in $\mathsf{\Omega }$:

$$\mathcal{H}\left(e\right)=\{\mathbf{v}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right):\mathrm{div}\mathbf{v}=0,\mathrm{rot}\mathbf{v}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\mathbf{v}\times \mathbf{n}{|}_{\mathsf{\Gamma }}=0\},$$

(7)

$$\mathcal{H}\left(m\right)=\{\mathbf{v}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right):\mathrm{div}\mathbf{v}=0,\mathrm{rot}\mathbf{v}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\mathbf{v}·\mathbf{n}{|}_{\mathsf{\Gamma }}=0\}.$$

(8)

The spaces $\mathcal{H}\left(e\right)$ and $\mathcal{H}\left(m\right)$ are finite-dimensional (see, e.g., Alekseev ([14], Chapter 6) and paper by Valli [70]), and $\dim \mathcal{H}\left(e\right)=p_0$, $\dim \mathcal{H}\left(m\right)=q_0$, where the numbers $p_0$ and $q_0$ are defined in Hypothesis 2.

Let $\mathcal{D}\left(\mathsf{\Omega }\right)$ be the space of infinitely differentiable finite functions in $\mathsf{\Omega }$, $H_0^1\left(\mathsf{\Omega }\right)$ is the completion of $\mathcal{D}\left(\mathsf{\Omega }\right)$ in $H^1\left(\mathsf{\Omega }\right),{\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)=H_0^1{\left(\mathsf{\Omega }\right)}^3$. The following is presumed: $\mathcal{T}=\{\phi \in H^1\left(\mathsf{\Omega }\right):\phi {|}_{{\mathsf{\Gamma }}_D}=0\}$, $\mathbf{V}=\{\mathbf{v}\in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right):\mathrm{div}\mathbf{v}=0\}$, ${\mathbf{L}}_T^2\left(\mathsf{\Gamma }\right)=\{\mathbf{h}\in {\mathbf{L}}^2\left(\mathsf{\Gamma }\right):\mathbf{h}·\mathbf{n}{|}_{\mathsf{\Gamma }}=0\}$, ${\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)=\{\mathbf{h}\in {\mathbf{H}}^{1/2}\left(\mathsf{\Gamma }\right):\mathbf{h}·\mathbf{n}{|}_{\mathsf{\Gamma }}=0\}$, $L_0^2\left(\mathsf{\Omega }\right)=\{p\in L^2\left(\mathsf{\Omega }\right):(p,1)=0\}$, $L_{+}^2\left({\mathsf{\Gamma }}_N\right)=\{\alpha \in L^2\left({\mathsf{\Gamma }}_N\right):\alpha \ge 0\}$, ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)=\{{\mathbf{v}\in \mathbf{H}}^1\left(\mathsf{\Omega }\right):\mathbf{v}·\mathbf{n}{|}_{\mathsf{\Gamma }}=0\}$, ${\mathbf{H}}_T^1(\mathsf{\Omega },\mathrm{div})=\{\mathbf{v}\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right):\mathrm{div}\mathbf{v}=0\}$, ${\mathbf{V}}_T=\{\mathbf{h}\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right):\mathrm{div}\mathbf{h}=0\}\cap \mathcal{H}{\left(m\right)}^{\perp }$. Spaces $\mathcal{T}$, $H_0^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$, $\mathbf{V}$, ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}_T^1(\mathsf{\Omega },\mathrm{div})$ and ${\mathbf{V}}_T$ are Hilbertian with norm ${\parallel ·\parallel }_{1,\mathsf{\Omega }}$. Along with spaces $H^1\left(\mathsf{\Omega }\right)$, $\mathcal{T}$, $H_0^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$, $\mathbf{V}$, ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$, ${\mathbf{V}}_T$ and ${\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)$ we will use dual of them spaces $H^1{\left(\mathsf{\Omega }\right)}^{*}$, ${\mathcal{T}}^{*}$, $H^{-1}\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}^{-1}\left(\mathsf{\Omega }\right)$, ${\mathbf{V}}^{*}$, ${\mathbf{H}}_T^{-1}\left(\mathsf{\Omega }\right)$, ${\mathbf{V}}_T^{*}$ and ${\mathbf{H}}_T^{-1/2}\left(\mathsf{\Gamma }\right)$ (see the book by Alekseev ([14], Sections 4.1 and 6.1) for more details on the properties of these spaces). Let us put

$${\tilde{H}}^s\left(\mathsf{\Gamma }\right):=\{h\in H^s\left(\mathsf{\Gamma }\right):{(h,1)}_{\mathsf{\Gamma }}=0\},0\le s\le 1/2.$$

By $\gamma$ or ${\gamma |}_{{\mathsf{\Gamma }}_D}$ we denote the trace operators acting from the space $H^1\left(\mathsf{\Omega }\right)$ to the trace space $H^{1/2}\left(\mathsf{\Gamma }\right)$ or to the trace space $H^{1/2}\left({\mathsf{\Gamma }}_D\right)$ of functions defined on the segment ${\mathsf{\Gamma }}_D$. It is well known that the above operators are continuous and have continuous right inverse operators ${\gamma }_r^{-1}$ and ${\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}$. For any function $\psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right)$, the function ${\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}\psi \in H^1\left(\mathsf{\Omega }\right)$ is often called the standard continuation (or replenishment) into $\mathsf{\Omega }$ of the boundary function $\psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right)$. For this function the following estimate is valid:

$$\parallel {\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}{\psi \parallel }_{1,\mathsf{\Omega }}\le C_D{\parallel \psi \parallel }_{1/2,\mathsf{\Gamma }}\forall \psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right),C_D=\parallel {\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}\parallel .$$

(9)

Along with the trace operators $\gamma$ and ${\gamma |}_{{\mathsf{\Gamma }}_D}$ we will also use the continuous normal trace operator

$${\gamma }_n:{\mathbf{H}}^r\left(\mathsf{\Omega }\right)\to H^s\left(\mathsf{\Gamma }\right),s=r-1/2,1/2\le r\le 1,$$

which puts in correspondence to each vector $\mathbf{h}\in {\mathbf{H}}^r\left(\mathsf{\Omega }\right)$ its normal trace $h_n\equiv \mathbf{h}·\mathbf{n}\in H^s\left(\mathsf{\Gamma }\right)$. The above operator is continuous and has a continuous right inverse ${\gamma }_n^{-1}:H^s\left(\mathsf{\Gamma }\right)\to {\mathbf{H}}^r\left(\mathsf{\Omega }\right)$, and the next estimate is valid:

$$\parallel {\gamma }_n^{-1}{q\parallel }_{r,\mathsf{\Omega }}\le C_n{\parallel q\parallel }_{H^s\left(\mathsf{\Gamma }\right)}\forall q\in H^s\left(\mathsf{\Gamma }\right).$$

(10)

One can read more about its properties in the books, respectively, by Alekseev ([14], Chapter 1) and by Girault et al. ([71], Chapter 1).

In the future, we will use the following Green’s formulas (see [14,71] for more details):

$${\int }_{\mathsf{\Omega }}\mathsf{\Delta }uvd\mathsf{\Omega }=-{\int }_{\mathsf{\Omega }}\nabla u·\nabla vd\mathsf{\Omega }+{\int }_{\partial \mathsf{\Omega }}\frac{\partial u}{\partial n}vd\sigma \forall u\in H^2\left(\mathsf{\Omega }\right),v\in H^1\left(\mathsf{\Omega }\right),$$

(11)

$${\int }_{\mathsf{\Omega }}\mathbf{v}·\mathrm{grad}\phi d\mathsf{\Omega }+{\int }_{\mathsf{\Omega }}\mathrm{div}\mathbf{v}\phi d\mathsf{\Omega }={\int }_{\partial \mathsf{\Omega }}\mathbf{v}·\mathbf{n}\phi d\sigma \forall \mathbf{v}\in H^1{\left(\mathsf{\Omega }\right)}^3,\phi \in H^1\left(\mathsf{\Omega }\right),$$

(12)

$${\int }_{\mathsf{\Omega }}(\mathbf{v}·\mathrm{rot}\mathbf{w}-\mathbf{w}·\mathrm{rot}\mathbf{v})d\mathsf{\Omega }={\int }_{\partial \mathsf{\Omega }}(\mathbf{v}\times \mathbf{n})·{\mathbf{w}}_{T}d\sigma \forall \mathbf{v},\mathbf{w}\in H^1{\left(\mathsf{\Omega }\right)}^3.$$

(13)

We define a series of bilinear forms associated with summands in the model Equations (1) and (2):

$$a_0(\mathbf{u},\mathbf{v})=(\nabla \mathbf{u},\nabla \mathbf{v})\equiv {\int }_{\mathsf{\Omega }}\nabla \mathbf{u}·\nabla \mathbf{v}d\mathsf{\Omega },a_1(\mathbf{H},\mathbf{\Psi })=(\mathrm{rot}\mathbf{H},\mathrm{rot}\mathbf{\Psi })\equiv {\int }_{\mathsf{\Omega }}\mathrm{rot}\mathbf{H}·\mathrm{rot}\mathbf{\Psi }d\mathsf{\Omega },$$

$$b(\mathbf{v},r)=-(\mathrm{div}\mathbf{v},r)\equiv -{\int }_{\mathsf{\Omega }}\mathrm{div}\mathbf{v}rd\mathsf{\Omega },\tilde{a}(T,S)=(\nabla T,\nabla S)\equiv {\int }_{\mathsf{\Omega }}\nabla T·\nabla Sd\mathsf{\Omega },$$

$$b_1(T,\mathbf{v})=(\mathbf{b},\mathbf{v}T)={\int }_{\mathsf{\Omega }}\mathbf{b}·\mathbf{v}Td\mathsf{\Omega },(\mathbf{b}=b\mathbf{G}).$$

(14)

We will use below the following inequalities:

$${\parallel \phi \parallel }_{L^s\left(\mathsf{\Omega }\right)}\le C_s{\parallel \phi \parallel }_{1,\mathsf{\Omega }}\forall \phi \in H^1\left(\mathsf{\Omega }\right),1\le s\le 6,$$

(15)

$${\parallel \mathbf{u}\parallel }_{L^s{\left(\mathsf{\Omega }\right)}^3}\le C_s{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\forall \mathbf{u}\in H^1{\left(\mathsf{\Omega }\right)}^3,1\le s\le 6,$$

(16)

$$|(\phi \mathbf{q},\mathbf{v})|\le C^p{\parallel \mathbf{q}\parallel }_{L^p\left(\mathsf{\Omega }\right)}{\parallel \phi \parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}\forall \mathbf{q}\in L^p{\left(\mathsf{\Omega }\right)}^3,\phi \in H^1\left(\mathsf{\Omega }\right),\mathbf{v}\in H^1{\left(\mathsf{\Omega }\right)}^3.$$

(17)

Here, $C_s$ is a positive constant depending on $\mathsf{\Omega }$ and on $s\in [1,6]$, and $C^p$ is a positive constant depending on $\mathsf{\Omega }$ and on p at $p\ge 3/2$. The inequalities (15) and (16) are consequences of Sobolev’s embedding theorem, according to which the space $H^1\left(\mathsf{\Omega }\right)$ is embedded into $L^s\left(\mathsf{\Omega }\right)$ continuously at $s\le 6$ and compactly at $s<6$.

In addition, we will use a number of properties of the bilinear and trilinear forms under consideration. Let us formulate them in the form of the following Lemma 1 (details of the proof can be found in the books [14,71]).

Lemma 1.

If Hypothesises 1 and 2 are satisfied, there are positive constants ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, β, ${\beta }_1$, ${\gamma }_0^{\prime }$, ${\gamma }_0$, ${\gamma }_1^{\prime }$, ${\gamma }_1$, ${\gamma }_2^{\prime }$, ${\gamma }_2$, ${\gamma }_3$, ${\gamma }_4$, $C_1$, $C_e$, $C_N$, depending on Ω such that the following relations hold:

$$((\mathbf{u}·\nabla )\mathbf{v},\mathbf{w})=-(\mathbf{u}·\nabla )\mathbf{w},\mathbf{v})\forall \mathbf{u}\in {\mathbf{H}}_{\mathrm{div}}^1\left(\mathsf{\Omega }\right),(\mathbf{v},\mathbf{w})\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right)\times {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right),$$

(18)

$$(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})=-(\mathrm{rot}\mathbf{\Psi }\times \mathbf{u},\mathbf{H})\forall (\mathbf{\Psi },\mathbf{H},\mathbf{u})\in {\mathbf{H}}^1{\left(\mathsf{\Omega }\right)}^3,$$

(19)

$$(\mathbf{u}·\nabla T,S)=-(\mathbf{u}·\nabla S,T)\forall \mathbf{u}\in {\mathbf{H}}_T^1(\mathsf{\Omega },\mathrm{div}),\forall T\in H^1\left(\mathsf{\Omega }\right),S\in \mathcal{T},$$

(20)

$$|a_0{(\mathbf{u},\mathbf{v})|\le \parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }},a_0(\mathbf{v},\mathbf{v})\ge {\alpha }_0{\parallel \mathbf{v}\parallel }^2\forall \left(\mathbf{u},\mathbf{v}\right)\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right)\times {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right),$$

(21)

$$|a_1(\mathbf{H},\mathbf{\Psi })|\le C_1^2{\parallel \mathbf{H}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }},$$

$$a_1(\mathbf{\Psi },\mathbf{\Psi })\equiv (\mathrm{rot}\mathbf{\Psi },\mathrm{rot}\mathbf{\Psi })\ge {\alpha }_1{\parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}^2\forall \mathbf{H}\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right),\mathbf{\Psi }\in {\mathbf{V}}_T,$$

(22)

$$|\tilde{a}{(T,S)|\le \parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel S\parallel }_{1,\mathsf{\Omega }},\tilde{a}(S,S)\ge {\alpha }_2{\parallel S\parallel }_{1,\mathsf{\Omega }}^2\forall T\in H^1\left(\mathsf{\Omega }\right),S\in \mathcal{T},$$

(23)

$$|\left((\mathbf{u}·\nabla )\mathbf{v},\mathbf{w}\right)|\le {\gamma }_0^{\prime }{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{w}\parallel }_{{\mathbf{L}}^4\left(\mathsf{\Omega }\right)}\le$$

$$\le {\gamma }_0{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{w}\parallel }_{1,\mathsf{\Omega }},{\gamma }_0=C_4{\gamma }_0^{\prime },\forall (\mathbf{u},\mathbf{v},\mathbf{w})\in {\mathbf{H}}^1{\left(\mathsf{\Omega }\right)}^3,$$

(24)

$$|(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})|\le {\gamma }_1^{\prime }{\parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{H}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{u}\parallel }_{{\mathbf{L}}^4\left(\mathsf{\Omega }\right)}\le$$

$$\le {\gamma }_1{\parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{H}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }},{\gamma }_1=C_4{\gamma }_1^{\prime },\forall (\mathbf{\Psi },\mathbf{H},\mathbf{u})\in {\mathbf{H}}^1{\left(\mathsf{\Omega }\right)}^3$$

(25)

$$\underset{0\ne \mathbf{v}\in H_0^1{\left(\mathsf{\Omega }\right)}^3}{sup}\frac{b(\mathbf{v},p)}{{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}}\ge \beta {\parallel p\parallel }_{\mathsf{\Omega }}\forall p\in L_0^2\left(\mathsf{\Omega }\right),$$

(26)

$$|(\mathbf{u}·\nabla T,S)|\le {\gamma }_2^{\prime }{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel S\parallel }_{L^4\left(\mathsf{\Omega }\right)}\le$$

$$\le {\gamma }_2{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel S\parallel }_{1,\mathsf{\Omega }},{\gamma }_2=C_4{\gamma }_2^{\prime },\forall \mathbf{u}\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right),(T,S)\in H^1\left(\mathsf{\Omega }\right)\times \mathcal{T}.$$

(27)

$$|b_1(T,\mathbf{v})|\le {\beta }_1{\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}\forall T\in H^1\left(\mathsf{\Omega }\right),{\mathbf{v}\in \mathbf{H}}_0^1\left(\mathsf{\Omega }\right).$$

(28)

$${|(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}|\le C_e{\parallel \mathbf{k}\parallel }_{\mathsf{\Gamma }}{\parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}\forall \mathbf{k}\in {\mathbf{L}}^2\left(\mathsf{\Gamma }\right),\mathbf{\Psi }\in {\mathbf{V}}_T,$$

(29)

$$|\langle f,S\rangle |\le {\gamma }_3{\parallel f\parallel }_{{\mathcal{T}}^{*}}{\parallel S\parallel }_{1,\mathsf{\Omega }},\forall f\in {\mathcal{T}}^{*},S\in \mathcal{T},$$

(30)

$${|(\chi ,S)}_{{\mathsf{\Gamma }}_N}|\le {\gamma }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}{\parallel S\parallel }_{1,\mathsf{\Omega }},\forall \chi \in L^2\left({\mathsf{\Gamma }}_N\right),S\in \mathcal{T},$$

(31)

$${|(\alpha T,S)}_{{\mathsf{\Gamma }}_N}|\le C_N{\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}{\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel S\parallel }_{1,\mathsf{\Omega }}\forall \alpha \in L_{+}^2\left({\mathsf{\Gamma }}_N\right),T\in H^1\left(\mathsf{\Omega }\right),S\in \mathcal{T}.$$

(32)

Here, $C_4$ is the constant included to (24) at $s=4$, ${\beta }_1=C_4^2{\parallel |\mathbf{b}\parallel }_{\mathsf{\Omega }}$.

The main role in the study of Problem 1 will be played, along with spaces $H^1\left(\mathsf{\Omega }\right)$, $H_0^1\left(\mathsf{\Omega }\right)$, $\mathcal{T}$, ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$, $\mathbf{V}$, ${\mathbf{V}}_T$, ${\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)$ and ${\tilde{H}}^s\left(\mathsf{\Gamma }\right)$, by the following function spaces

$${\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)=\{\mathbf{h}\in H_{DC}\left(\mathsf{\Omega }\right)\cap {\mathbf{H}}^r\left(\mathsf{\Omega }\right)\cap \mathcal{H}{\left(m\right)}^{\perp }:\mathrm{div}\mathbf{h}=0\}\supset {\mathbf{V}}_T,$$

(33)

$$H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })=\{\mathbf{e}\in H(\mathrm{curl},\mathsf{\Omega }):\mathrm{curl}\mathbf{e}=0,\mathbf{e}\times \mathbf{n}{|}_{\mathsf{\Gamma }}\in {\mathbf{L}}_T^2\left(\mathsf{\Gamma }\right)\},$$

(34)

$$H={\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right),H_{0T}={\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathbf{V}}_T,V_{0T}=\mathbf{V}\times {\mathbf{V}}_T\subset H_{0T}\subset H,$$

(35)

as well as dual of $H_{0T}$ and $V_{0T}$ spaces $H_{0T}^{*}={\mathbf{H}}^{-1}\left(\mathsf{\Omega }\right)\times {\mathbf{V}}_T^{*}$ and $V_{0T}^{*}={\mathbf{V}}^{*}\times {\mathbf{V}}_T^{*}$. The spaces ${\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$ and $H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })$ are Hilbert spaces with norms, respectively:

$${\parallel \mathbf{h}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}={\parallel \mathbf{h}\parallel }_{r,\mathsf{\Omega }}+{\parallel \mathrm{rot}\mathbf{h}\parallel }_{\mathsf{\Omega }},1/2\le r\le {1,\parallel \mathbf{e}\parallel }_{H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })}={\parallel \mathbf{e}\parallel }_{\mathsf{\Omega }}+{\parallel \mathbf{e}\times \mathbf{n}\parallel }_{\mathsf{\Gamma }}.$$

The spaces $H_{0T}$ and $V_{0T}$ are Hilbert spaces with a Hilbert norm

$$(\mathbf{u},\mathbf{H})\to {\parallel (\mathbf{u},\mathbf{H})\parallel }_{1,\mathsf{\Omega }}\equiv {(\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2+{\kappa \parallel \mathbf{H}\parallel }_{1,\mathsf{\Omega }}^2{)}^{1/2},$$

(36)

where the parameter $\kappa =\mu {\rho }_0^{-1}$ was defined in (4). The goal of introducing the multiplier $\kappa$ into the norm (36) is to equalize the dimensions of both summands ${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2$ and ${\parallel \mathbf{H}\parallel }_{1,\mathsf{\Omega }}^2$ in the right-hand side of (36) (see Alekseev ([14], p. 283)).

In order to prove this fact we will assume that the norms ${\parallel ·\parallel }_{\mathsf{\Omega }}$ and ${\parallel ·\parallel }_{1,\mathsf{\Omega }}$ of a function u in $L^2\left(\mathsf{\Omega }\right)$ and in $H^1\left(\mathsf{\Omega }\right)$ and seminorm ${|·|}_{1,\mathsf{\Omega }}$ in $H^1\left(\mathsf{\Omega }\right)$ are defined as follows:

$${\parallel u\parallel }_{\mathsf{\Omega }}^2={\int }_{\mathsf{\Omega }}{u}^2{d\mathsf{\Omega },|u|}_{1,\mathsf{\Omega }}^2={\int }_{\mathsf{\Omega }}{|\nabla u|}^2{d\mathsf{\Omega },\parallel u\parallel }_{1,\mathsf{\Omega }}^2=l^{-2}{\parallel u\parallel }_{\mathsf{\Omega }}^2+{|u|}_{1,\mathsf{\Omega }}^2,\left[l\right]=L_0.$$

(37)

Here, l is the dimensional factor of dimension $\left[l\right]=L_0$ whose value is equal to 1 (see Formula (6)). Using (37) it is easy to verify that the dimensions of ${\parallel u\parallel }_{\mathsf{\Omega }},{|u|}_{1,\mathsf{\Omega }}$ and ${\parallel u\parallel }_{1,\mathsf{\Omega }}$ are related to the dimension $\left[u\right]$ of u by formulas

$${[\parallel u\parallel }_{\mathsf{\Omega }}]=\left[u\right]L_0^{3/2}{and[|u|}_{1,\mathsf{\Omega }}{]=[\parallel u\parallel }_{1,\mathsf{\Omega }}]=\left[u\right]L_0^{1/2}.$$

We recall also (see, e.g., Alekseev ([14], Chapters 2, 6)) that

$$\left[\mathbf{u}\right]=\left[\mathbf{g}\right]=L_0/T_0,\left[\mathbf{H}\right]=\left[q\right]=I_0/L_0,\left[\mathbf{j}\right]=I_0/L_0^2,$$

$$\left[T\right]=\left[\psi \right]=K,\left[\chi \right]=K_0{L}_0/T_0,\left[{\nu }_m\right]=\left[\nu \right]=\left[\lambda \right]=L_0^2/T_0,\left[\kappa \right]=L_0^4/T_0^2{I}_0^2.$$

(38)

Here, and below $T_0$, $M_0$, $K_0$, and $I_0$ denote the SI dimensions of time, mass, temperature and current expressed in units of second, kilogram, kelvin and ampere, respectively.

Using (21)–(25), (26), (37) and (38) one can easily derive that

$$\left[{\alpha }_i\right]=1,\left[{\gamma }_i\right]=L_0^{1/2},i=0,1,2,\left[\beta \right]=1,\left[{\beta }_1\right]=L_0^3/T_0^2{K}_0,$$

$$\left[M_{\mathbf{u}}^0\right]={[\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{]=[\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}]=L_0^{3/2}/T_0,\left[M_p^0\right]=L_0^{7/2}/T_0^2,$$

$$\left[M_T^0\right]={[\parallel T\parallel }_{1,\mathsf{\Omega }}{]=[\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}]=K_0{L}_0^{1/2},$$

$$\left[M_{\mathbf{H}}^0\right]={[\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}{]=[\parallel q\parallel }_{s,\mathsf{\Gamma }}{]=[\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}]=I_0/L_0^{1/2}.$$

(39)

In particular, from (39) follows that

$$[\sqrt{\kappa }{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}]=L_0^{3/2}/T_0={[\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}].$$

Let us note that the spaces ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$ (or ${\mathbf{H}}_T^1(\mathsf{\Omega },\mathrm{div})$), ${\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$, $H^1\left(\mathsf{\Omega }\right)$, $H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })$ and $L_0^2\left(\mathsf{\Omega }\right)$ defined above will play the role of solution spaces, respectively, for velocity, magnetic field, temperature, electric field and pressure, while the spaces ${\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$ (or $\mathbf{V}$), ${\mathbf{V}}_T$ and $\mathcal{T}$ will serve as the corresponding test function spaces for velocity, magnetic field and temperature, respectively. We also remark that the space ${\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$ transforms at $r=1$ to the space:

$${\mathcal{H}}_m^1\left(\mathsf{\Omega }\right)=\{\mathbf{h}\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right)\cap \mathcal{H}{\left(m\right)}^{\perp }:\mathrm{div}\mathbf{h}=0\},$$

(40)

which plays the role of the solution space for the magnetic field $\mathbf{H}$ in Alekseev [13]. It is clear that the following chain of embeddings is valid:

$${\mathbf{V}}_T\subset {\mathcal{H}}_m^1\left(\mathsf{\Omega }\right)\subseteq {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right),1/2\le r\le 1.$$

Along with the inequality (25) we will also use a more general inequality

$$|({\mathbf{H}}_1\times {\mathbf{H}}_2,\mathbf{u})|\le {\gamma }_1\parallel {\mathbf{H}}_1{\parallel }_{{\mathcal{H}}_1}\parallel {\mathbf{H}}_2{\parallel }_{{\mathcal{H}}_2}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\forall {\mathbf{H}}_1\in {\mathcal{H}}_1,{\mathbf{H}}_2\in {\mathcal{H}}_2,\mathbf{u}\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right),$$

(41)

where each of the spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ coincides with one of the spaces ${\mathbf{H}}^1\left(\mathsf{\Omega }\right)$ or ${\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$ at $1/2\le r\le 1$.

Let us put ${\nu }_{*}=min({\alpha }_0\nu ,{\alpha }_1{\nu }_m)$ and define bilinear forms:

$$a_2(T,S)=\lambda \tilde{a}(T,S)+\lambda {(\alpha T,S)}_{{\mathsf{\Gamma }}_N},$$

(42)

$$a((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))=\nu a_0(\mathbf{u},\mathbf{v})+{\nu }_1{a}_1(\mathbf{H},\mathbf{\Psi })\equiv \nu a_0(\mathbf{u},\mathbf{v})+\kappa {\nu }_m{a}_1(\mathbf{H},\mathbf{\Psi }),$$

(43)

where $\alpha \in L_{+}^2\left({\mathsf{\Gamma }}_N\right)$ is a given function, ${\nu }_m={\nu }_1/\kappa$ is the magnetic viscosity. By virtue of (21)–(23) and (32), the form $a_2$ is continuous on $H^1\left(\mathsf{\Omega }\right)$ and coercive on $\mathcal{T}$, and the form a is continuous on ${\mathbf{H}}^1{\left(\mathsf{\Omega }\right)}^2$ and coercive on $H_{0T}$. This means that

$$|a_2(T,S)|\le \lambda (1+C_N{\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}{)\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel S\parallel }_{1,\mathsf{\Omega }},a_2(S,S)\ge {\alpha }_2\lambda {\parallel S\parallel }_{1,\mathsf{\Omega }}^2\forall S\in \mathcal{T},$$

(44)

$$a(\left(\mathbf{v},\mathbf{\Psi }\right),\left(\mathbf{v},\mathbf{\Psi }\right))\ge {\nu }_{*}{\parallel \left(\mathbf{v},\mathbf{\Psi }\right)\parallel }_{1,\mathsf{\Omega }}^2\equiv {\nu }_{*}{(\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}^2+{\kappa \parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}^2)\forall \left(\mathbf{v},\mathbf{\Psi }\right)\in H_{0T}.$$

(45)

Let the bilinear continuous form $\widehat{a}:H_{0T}\times H_{0T}\to \mathbb{R}$ satisfies the following “$\delta$-smallness” condition on $V_{0T}$:

$$|\widehat{a}{((\mathbf{v},\mathbf{\Psi }),(\mathbf{v},\mathbf{\Psi }))|\le \delta (\parallel \mathbf{v}\parallel }_{1,\mathsf{\Omega }}^2+{\kappa \parallel \mathbf{\Psi }\parallel }_{1,\mathsf{\Omega }}^2)\forall (\mathbf{v},\mathbf{\Psi })\in V_{0T},0\le \delta <{\nu }_{*}.$$

(46)

Let us consider for arbitrary elements $l\in {\mathcal{T}}^{*}$ and $\mathbf{F}\in V_{0T}^{*}$ variational problems consisting in finding such elements $T\in H^1\left(\mathsf{\Omega }\right)$ and $(\mathbf{u},\mathbf{H})\in V_{0T}$ that

$$a_2{(T,S)+(\mathbf{u}·\nabla T,S)=\langle l,S\rangle \forall S\in \mathcal{T},T|}_{{\mathsf{\Gamma }}_D}=\psi ,$$

(47)

$$a((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))+\widehat{a}((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))=\langle \mathbf{F},(\mathbf{v},\mathbf{\Psi })\rangle \forall (\mathbf{v},\mathbf{\Psi })\in V_{0T}.$$

(48)

The following two lemmas are proved using (44)–(46) and the Lax–Milgram theorem.

Lemma 2.

Let, under Hypothesises 1 and 2, the conditions $\alpha \in L_{+}^2\left({\mathsf{\Gamma }}_N\right)$, $\mathbf{u}\in {\mathbf{H}}_T^1(\mathsf{\Omega },\mathrm{div})$ be satisfied. Then: (1) the bilinear form $a_2(·,·)+(\mathbf{u}·\nabla ·,·)$ in (47) is continuous and coercive on $\mathcal{T}$ with constant ${\alpha }_2\lambda$; (2) for $\psi =0$ the problem (47) has a unique solution $T\in \mathcal{T}$ for any $l\in {\mathcal{T}}^{*}$ and the following estimate holds: ${\parallel T\parallel }_{1,\mathsf{\Omega }}\le (1/{\alpha }_2\lambda ){\parallel l\parallel }_{{\mathcal{T}}^{*}}$; (3) for any function $\psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right)$ there exists a unique solution $T\in H^1\left(\mathsf{\Omega }\right)$ to the problem (47) and for this solution the next estimate is valid:

$${\parallel T\parallel }_{1,\mathsf{\Omega }}\le (1/{\alpha }_2\lambda )\left[{\parallel l\parallel }_{{\mathcal{T}}^{*}}+({\alpha }_2\lambda +\lambda +\lambda C_N{\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}+{\gamma }_2{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }})C_D{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}\right],$$

(49)

where $C_D$ is the norm of the trace operator ${\gamma |}_{{\mathsf{\Gamma }}_D}:H^1\left(\mathsf{\Omega }\right)\to H^{1/2}\left({\mathsf{\Gamma }}_D\right)$ defined in (9).

Proof of Lemma 2.

Assertion (1) is a consequence of the inequalities (44) and of the identity $(\mathbf{u}·\nabla S,S)=0$ for all $S\in \mathcal{T}$ following from (20). Assertion (2), in which the solution T and the test function S are chosen from the same space $\mathcal{T}$, is a direct consequence of the statement (1) and of the Lax–Milgram theorem. To prove the assertion (3) of Lemma 2, the solution T of the problem (47) should be represented as $T=T_0+\tilde{T}$, where $T_0={\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}\psi$ is the standard continuation of the function $\psi$ inside the domain $\mathsf{\Omega }$, and $\tilde{T}\in \mathcal{T}$ is the new unknown function. Subsequent substituting the above representation into (47) leads to the homogeneous analog of the problem (47) with respect to $\tilde{T}\in \mathcal{T}$ corresponding to the case $\psi =0$. In this case, the estimate (49) is a consequence of Assertion 2 applied to the above homogeneous analog. □

According to a similar scheme, one can prove the following:

Lemma 3.

Let the bilinear continuous forms a and $\widehat{a}$ satisfy the conditions (45) and (46) when Hypothesises 1 and 2 are satisfied. Then: (1) the bilinear form $a+\widehat{a}$ is continuous and coercive on $V_{0T}$; (2) the problem (48) has a unique solution $(\mathbf{u},\mathbf{H})\in V_{0T}$ for any element $\mathbf{F}\in V_{0T}^{*}$ and the estimate ${\parallel (\mathbf{u},\mathbf{H})\parallel }_{1,\mathsf{\Omega }}\le {({\nu }_{*}-\delta )}^{-1}{\parallel \mathbf{F}\parallel }_{V_{0T}^{*}}$ is valid.

3. Solvability of Problem 1 and of Its Linear Analogue

Let us recall that our goal is to analyze the solvability, uniqueness and stability of solutions of the control problems for the MHDB model under consideration. Our analysis will be based on the theory of smooth-convex extremum problems in Hilbert spaces. As applied to models of hydrodynamics and magnetic hydrodynamics, this theory is based on the use of a weak formulation of the boundary value problem under consideration as a conditional constraint for the main state and for quantities that play the role of controls in the control problems under study. Therefore, we begin this section with the formulation and proof of some facts concerning the existence and uniqueness of the weak solution of Problem 1 and of its generalized linear analog.

In addition to Hypothesises 1 and 2, we assume that the following hypothesises are satisfied:

Hypothesis 3.

$\mathbf{f}\in {\mathbf{H}}^{-1}\left(\mathsf{\Omega }\right)$, $f\in {\mathcal{T}}^{*}$, $\mathbf{b}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$, $\mathbf{k}\in {\mathbf{L}}_T^2\left(\mathsf{\Gamma }\right)$, $\alpha \in L_{+}^2\left({\mathsf{\Gamma }}_N\right)$;

Hypothesis 4.

$\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$;

Hypothesis 5.

${\mathbf{g}\in \mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)$, $q\in {\tilde{H}}^s\left(\mathsf{\Gamma }\right)$, $0\le s\le 1/2$, $\psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right)$, $\chi \in L^2\left({\mathsf{\Gamma }}_N\right)$.

Let us define functionals $l:\mathcal{T}\to \mathbb{R}$ and $\mathbf{F}:H_{0T}\to \mathbb{R}$ by the formulas

$$\langle l,S\rangle =\langle f,S\rangle +{(\chi ,S)}_{{\mathsf{\Gamma }}_N},\langle \mathbf{F},\left(\mathbf{v},\mathbf{\Psi }\right)\rangle =\langle \mathbf{f},\mathbf{v}\rangle +{\nu }_1(\mathbf{j},\mathrm{rot}\mathbf{\Psi })+{(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}.$$

(50)

It follows from Hypothesises 3–5 and (22), (29), (30), (31), that $l\in {\mathcal{T}}^{*},\mathbf{F}\in H_{0T}^{*}$ and that

$${\parallel l\parallel }_{{\mathcal{T}}^{*}}\le {\gamma }_3{\parallel f\parallel }_{{\mathcal{T}}^{*}}+{\gamma }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N},$$

$${\parallel \mathbf{F}\parallel }_{H_{0T}^{*}}\le M={\parallel \mathbf{f}\parallel }_{-1,\mathsf{\Omega }}+C_1{\nu }_1{\kappa }^{-1/2}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+C_e{\parallel \mathbf{k}\parallel }_{\mathsf{\Gamma }}.$$

(51)

Following the variational method of investigating boundary value problems, we will define a weak formulation of Problem 1. To this end, we multiply the first equation in (1) by the function $\mathbf{v}\in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$, the first equation in (2) by $\mathrm{rot}\mathbf{\Psi }$ where $\mathbf{\Psi }\in {\mathbf{V}}_T$, and integrate over $\mathsf{\Omega }$. Using the Green’s formulas (11)–(13), the equation $\mathrm{rot}\mathbf{E}=0$ in (2) for the electric field $\mathbf{E}$ and the identity $(\mathbf{E},\mathrm{rot}\mathbf{\Psi })={(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}$ for $\mathbf{E}\in H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })$, $\mathbf{\Psi }\in {\mathbf{V}}_T$ and $\mathbf{k}\in {\mathbf{L}}_T^2\left(\mathsf{\Gamma }\right)$, we obtain, taking into account the notations of Section 2, that

$$\nu a_0(\mathbf{u},\mathbf{v})+((\mathbf{u}·\nabla )\mathbf{u},\mathbf{v})-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},\mathbf{v})+b(\mathbf{v},p)+b_1(T,\mathbf{v})=\langle \mathbf{f},\mathbf{v}\rangle \forall \mathbf{v}\in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right),$$

(52)

$${\nu }_1{a}_1(\mathbf{H},\mathbf{\Psi })+\kappa (\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})={\nu }_1(\mathbf{j},\mathrm{rot}\mathbf{\Psi })+{(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}\forall \mathbf{\Psi }\in {\mathbf{V}}_T.$$

(53)

Adding (52) and (53), we have

$$a\left(\right(\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }\left)\right)+\left(\right(\mathbf{u}·\nabla \left)\mathbf{u},\mathbf{v}\right)+\kappa \left[\right(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})-(\mathrm{rot}\mathbf{H}\times \mathbf{H},\mathbf{v}\left)\right]+$$

$$+b(\mathbf{v},p)+b_1(T,\mathbf{v})=\langle \mathbf{F},\left(\mathbf{v},\mathbf{\Psi }\right)\rangle \forall \left(\mathbf{v},\mathbf{\Psi }\right)\in H_{0T}.$$

(54)

Similarly, let us multiply the Equation (3) by $S\in \mathcal{T}$ and integrate over $\mathsf{\Omega }$. Using Green’s formula (11), we obtain

$$a_2(T,S)+(\mathbf{u}·\nabla T,S)\equiv \lambda \tilde{a}(T,S)+\lambda {(\alpha T,S)}_{{\mathsf{\Gamma }}_N}+(\mathbf{u}·\nabla T,S)=\langle l,S\rangle \forall S\in \mathcal{T}.$$

(55)

As a result, we obtained a weak formulation of Problem 1. It consists in finding the quadruple $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$, $1/2\le r\le 1$, satisfying the identities (54), (55) and the relations

$${\mathrm{div}\mathbf{u}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\mathbf{u}|}_{\mathsf{\Gamma }}=\mathbf{g},\mathbf{H}·\mathbf{n}{{|}_{\mathsf{\Gamma }}=q,T|}_{{\mathsf{\Gamma }}_D}=\psi .$$

(56)

The identity (54) does not contain the electric field $\mathbf{E}$. Nevertheless, arguing as in Alekseev [42], the field $\mathbf{E}\in H_{\mathsf{\Gamma }}^0(\mathrm{curl},\mathsf{\Omega })$ can be recovered from the quadruple $(\mathbf{u},\mathbf{H},T,p)$$\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$ satisfying (54), such that the Equations (2) are satisfied almost everywhere in $\mathsf{\Omega }$, and the first equation in (1) and the Equation (3) are satisfied in the sense of generalized functions. This allows us to set correctly the following definition.

Definition 1.

The weak solution of Problem 1 is any quadruple $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$, $1/2\le r\le 1$ satisfying (54)–(56).

Let the above quadruple $(\mathbf{u},\mathbf{H},p,T)$ be a solution to the problem (54)–(56). Considering the restriction of the identity (54) to the space $V_{0T}\equiv \mathbf{V}\times {\mathbf{V}}_T\subset H_{0T}$, we conclude that the triple $(\mathbf{u},\mathbf{H},T)$ satisfies the identity

$$a\left(\right(\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }\left)\right)+\left(\right(\mathbf{u}·\nabla )\mathbf{u},\mathbf{v})+\kappa \left[\right(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})-(\mathrm{rot}\mathbf{H}\times \mathbf{H},\mathbf{v}\left)\right]+$$

$$+b_1(T,\mathbf{v})=\langle \mathbf{F},(\mathbf{v},\mathbf{\Psi })\rangle \forall (\mathbf{v},\mathbf{\Psi })\in V_{0T}.$$

(57)

It is well known that the presence of inhomogeneous boundary conditions, such as the boundary conditions in (4) and (5) for Problem 1, significantly complicates its theoretical analysis. The standard method of investigating this type of problem is based on the use of so-called liftings, i.e., certain extensions inside the domain of $\mathsf{\Omega }$ of boundary data having special properties. In our case, the existence of liftings is provided by the following Lemma (see details of its proof in [13,72,73]).

Lemma 4.

Let Hypothesises 1 and 2 be satisfied. Then:

(1) For any function $\mathbf{g}\in {\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)$ and any number $\epsilon >0$, there exists such a vector ${\mathbf{u}}_{\epsilon }\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$ (velocity lifting) satisfying the following relations:

$$\mathrm{div}{\mathbf{u}}_{\epsilon }=0in\mathsf{\Omega },{\mathbf{u}}_{\epsilon }{|}_{\mathsf{\Gamma }}=\mathbf{g},$$

$$\parallel {\mathbf{u}}_{\epsilon }{\parallel }_{L^4\left(\mathsf{\Omega }\right)}\le {\epsilon \parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }},\parallel {\mathbf{u}}_{\epsilon }{\parallel }_{1,\mathsf{\Omega }}\le C_{\epsilon }{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}.$$

(58)

Here, $C_{\epsilon }$ is a constant depending on the parameter $\epsilon >0$ and the domain Ω.

(2) For any function $q\in {\tilde{H}}^s\left(\mathsf{\Gamma }\right)$, $0\le s\le 1/2$, there exists a unique function (magnetic lifting) ${\mathbf{H}}_0\in {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$, where $r=s+1/2$, such that

$$\mathrm{rot}{\mathbf{H}}_0=0in\mathsf{\Omega },{\mathbf{H}}_0{·\mathbf{n}|}_{\mathsf{\Gamma }}=q,\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le C_{\mathsf{\Gamma }}^s{\parallel q\parallel }_{s,\mathsf{\Gamma }}.$$

(59)

Here, $C_{\mathsf{\Gamma }}^s$ is a constant depending on s and Γ, but independent of a boundary function q.

(3) There exists a family of continuous non-decreasing functions $M_{\delta }:(0,\infty )\to (0,\infty )$ with $M_{\delta }\left(0\right)=0$, $\delta \in (0,1]$, depending on the parameter $\delta >0$ and of the domain Ω such that for any function $\psi \in H^{1/2}\left({\mathsf{\Gamma }}_D\right)$ there is a function $T_{\delta }\in H^1\left(\mathsf{\Omega }\right)$ (temperature lifting) that

$$T_{\delta }=\psi on{\mathsf{\Gamma }}_D,\parallel T_{\delta }{\parallel }_{L^4\left(\mathsf{\Omega }\right)}\le \delta ,\parallel T_{\delta }{\parallel }_{1,\mathsf{\Omega }}\le M_{\delta }{(\parallel \psi \parallel }_{1,2,{\mathsf{\Gamma }}_D}).$$

(60)

Now, we are able to formulate the following existence theorem for the weak solution to Problem 1.

Theorem 1.

When Hypothesises 1–5 are satisfied, there is a weak solution $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$ to Problem 1, where $r=s+1/2$, and the following estimates are valid for this solution:

$${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\le M_{\mathbf{u}}{,\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le M_{\mathbf{H}}{,\parallel T\parallel }_{1,\mathsf{\Omega }}\le M_T,{\parallel p\parallel }_{\mathsf{\Omega }}\le M_p.$$

(61)

Here, $M_{\mathbf{u}}$, $M_{\mathbf{H}}$, $M_T$, and $M_p$ are continuous non-decreasing functions of the following norms: ${\parallel \mathbf{f}\parallel }_{-1,\mathsf{\Omega }}$, ${\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}$, ${\parallel f\parallel }_{{\mathcal{T}}^{*}}$, ${\parallel \mathbf{b}\parallel }_{\mathsf{\Omega }}$, ${\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}$, ${\parallel q\parallel }_{s,\mathsf{\Gamma }}$, ${\parallel \mathbf{k}\parallel }_{\mathsf{\Gamma }}$, ${\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}$, ${\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}$ and ${\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}$. If, in addition, the data $\mathbf{f}$, $\mathbf{j}$, f, $\mathbf{b}$, $\mathbf{g}$, q, $\mathbf{k}$, ψ, χ and α are small (or the coefficients $\nu ,{\nu }_m$, and λ are large) in the sense that the following conditions are fulfilled:

$${\gamma }_0{M}_{\mathbf{u}}+{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}+\frac{{\gamma }_2{\beta }_1}{{\alpha }_2\lambda }{M}_T<{\alpha }_0\nu ,$$

(62)

$${\gamma }_1{M}_{\mathbf{u}}+{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}<{\alpha }_1{\nu }_m,$$

(63)

where the constants ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\gamma }_0$, ${\gamma }_1$, ${\gamma }_2$, and ${\beta }_1$ are defined in Lemma 1, then the weak solution is unique.

Proof of Theorem 1.

In the case where $\psi =0$ and $s=1/2$ and hence the temperature is sought in the space $\mathcal{T}$ and the magnetic component $\mathbf{H}$ is sought in the space ${\mathcal{H}}_m^1\left(\mathsf{\Omega }\right)$ defined in (40), Theorem 1 is proved in Alekseev [36]. The idea of the proof is to look for the components $\mathbf{u}$ and $\mathbf{H}$ of the weak solution in the form

$$\mathbf{u}={\mathbf{u}}_{\epsilon }+\tilde{\mathbf{u}},\mathbf{H}={\mathbf{H}}_0+\tilde{\mathbf{H}}.$$

(64)

Here, ${\mathbf{u}}_{\epsilon }$ and ${\mathbf{H}}_0$ are the hydrodynamic and magnetic liftings defined in Lemma 4, and $\tilde{\mathbf{u}}\in \mathbf{V}$ and $\tilde{\mathbf{H}}\in {\mathbf{V}}_T$ are the new unknown functions that we are looking for. Subsequent substitution of the relations (64) into (55)–(57) leads to a nonlinear system with respect to the quadruple $(\tilde{\mathbf{u}},\tilde{\mathbf{H}},T,p)$ depending on the parameter $\epsilon >0$. Then, using the lifting properties and Schauder fixed-point theorem, we prove that for small values of the parameter $\epsilon$ the obtained system has at least one solution ($\tilde{\mathbf{u}},\tilde{\mathbf{H}},T,p$) $\in \mathbf{V}\times {\mathbf{V}}_T\times \mathcal{T}\times L_0^2\left(\mathsf{\Omega }\right)$.

Moreover, for a given solution ($\tilde{\mathbf{u}},\tilde{\mathbf{H}},T,p$), we derive estimates of norms $\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}$, $\parallel \tilde{\mathbf{H}}{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}$, ${\parallel T\parallel }_{1,\mathsf{\Omega }}$ and ${\parallel p\parallel }_{\mathsf{\Omega }}$ via the data. From these estimates, in turn, we based on the representation (64) derive estimates of the form (61) for the quadruple $(\mathbf{u},\mathbf{H},T,p)$, which is the desired weak solution of Problem 1 at $\psi =0$ and $s=1/2$.

The existence of the solution of Problem 1 in the case when $\psi \ne 0$ and $s\in [0,1/2]$ is an arbitrary number is proved by a similar scheme using, in addition to hydrodynamic and magnetic liftings, the temperature lifting $T_{\delta }$ introduced in Lemma 4. Using this lifting the temperature T is sought in the form $T=T_{\delta }+\tilde{T}$, where $\tilde{T}\in \mathcal{T}$ is the new function. □

Along with the nonlinear Problem 1, we will consider a generalized linear analogue of Problem 1, which consists in finding the quadruple $(\mathbf{u},\mathbf{H},T,p)$ from the next linear relations:

$$-\nu \mathsf{\Delta }\mathbf{u}+({\mathbf{u}}^0·\nabla )\mathbf{u}+\nabla p-\kappa \mathrm{rot}\mathbf{H}\times {\mathbf{H}}^0=\mathbf{f}-\mathbf{b}T,\mathrm{div}\mathbf{u}=\sigma \mathrm{i}\mathrm{n}\mathsf{\Omega },$$

(65)

$${\nu }_1\mathrm{rot}\mathbf{H}-\mathbf{E}+\kappa {\mathbf{H}}^0\times \mathbf{u}={\nu }_1\mathbf{j},\mathrm{div}\mathbf{H}=0,\mathrm{rot}\mathbf{E}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },$$

(66)

$${\mathbf{u}|}_{\mathsf{\Gamma }}=\mathbf{g},\mathbf{H}·\mathbf{n}{{|}_{\mathsf{\Gamma }}=q,\mathbf{E}\times \mathbf{n}|}_{\mathsf{\Gamma }}=\mathbf{k},$$

(67)

$$-\lambda \mathsf{\Delta }T+{\mathbf{u}}^0·\nabla T=f\mathrm{i}\mathrm{n}\mathsf{\Omega },T=\psi \mathrm{o}\mathrm{n}{\mathsf{\Gamma }}_D,\lambda (\partial T/\partial n+\alpha T)=\chi \mathrm{o}\mathrm{n}{\mathsf{\Gamma }}_N.$$

(68)

Here, in addition to the known functions $\mathbf{f}$, $\mathbf{j}$, $\mathbf{b}$, f, $\mathbf{g}$, q, $\mathbf{k}$, $\psi$, $\chi$, $\alpha$, the following given functions are added: “velocity” ${\mathbf{u}}^0$, “magnetic field strength” ${\mathbf{H}}^0$ and function $\sigma$ satisfying the conditions:

Hypothesis 6.

${\mathbf{u}}^0\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}^0\in {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$, $1/2\le r\le 1$;

Hypothesis 7.

$\sigma \in L_0^2\left(\mathsf{\Omega }\right)$.

In fact, we have somewhat generalized the linear analog of Problem 1 by replacing the solenoidality condition $\mathrm{div}\mathbf{u}=0$ with the more general condition $\mathrm{div}\mathbf{u}=\sigma$. Furthermore, the elements $\mathbf{F}\in H_{0T}^{*}$ and $l\in {\mathcal{T}}^{*}$ below (in this Section) will denote arbitrary functionals in general, not necessarily coinciding with the functionals $\mathbf{F}$ and l defined in (50). The corresponding weak formulation of the problem (65)–(68) is to find a quadruple $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$ from the conditions

$$a((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))+(({\mathbf{u}}^0·\nabla )\mathbf{u},\mathbf{v})+\kappa [(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}^0,\mathbf{u})-(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}^0,\mathbf{v})]+b(\mathbf{v},p)=$$

$$=\langle \mathbf{F},(\mathbf{v},\mathbf{\Psi })\rangle -b_1(T,\mathbf{v})\forall (\mathbf{v},\mathbf{\Psi })\in H_{0T}\equiv {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathbf{V}}_T,$$

(69)

$${\mathrm{div}\mathbf{u}=\sigma ,\mathbf{u}|}_{\mathsf{\Gamma }}{=\mathbf{g},\mathbf{H}·\mathbf{n}|}_{\mathsf{\Gamma }}=q,$$

(70)

$$a_2(T,S)+({\mathbf{u}}^0·\nabla T,S)=\langle l,S\rangle {\forall S\in \mathcal{T},T|}_{{\mathsf{\Gamma }}_D}=\psi .$$

(71)

The linearization of Problem 1 that we use has two features. The first feature is that (71) is a $\mathbf{u},\mathbf{H}$- and p-independent problem for T with a bilinear form in the left-hand side of (71) which is coercive on $\mathcal{T}$, when the condition on $\alpha$ in Hypothesis 3 and the condition on ${\mathbf{u}}^0$ in Hypothesis 6 are satisfied. Therefore, by Lemma 2, its solution $T\in H^1\left(\mathsf{\Omega }\right)$ exists, is unique, and the next estimate for it holds:

$${\parallel T\parallel }_{1,\mathsf{\Omega }}\le {\widehat{M}}_T\equiv (1/{\alpha }_2\lambda )\times$$

$$\times \left[{\parallel l\parallel }_{{\mathcal{T}}^{*}}+({\alpha }_2\lambda +\lambda +\lambda C_N{\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}+{\gamma }_2\parallel {\mathbf{u}}^0{\parallel }_{1,\mathsf{\Omega }})C_D{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}\right].$$

(72)

The second feature is that the problem (69), (70) for a given function $T\in H^1\left(\mathsf{\Omega }\right)$ is a weak formulation of the linear MHD model studied in the case when $r=1$ in Alekseev [13]. The summands $(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}_0,\mathbf{u})$ and $-(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}_0,\mathbf{v})$ obtained by linearizing the summand ($\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u}$) corresponding to the Maxwell advective term $\mathbf{H}\times \mathbf{u}$ in the first equation of (1), and the summand $-(\mathrm{rot}\mathbf{H}\times \mathbf{H},\mathbf{v})$ from (52) corresponding to the Lorentzian force $\mathrm{rot}\mathbf{H}\times \mathbf{H}$ in (2), sum to annihilate at $\left(\mathbf{u},\mathbf{H}\right)=(\mathbf{v},\mathbf{\Psi })$. This together with (17) entails the coercivity on the space $H_{0T}$ of the bilinear with respect to $(\mathbf{u},\mathbf{H})$ and $(\mathbf{v},\mathbf{\Psi })$ form, standing in the left-hand side of (69). The latter gives the key to proving the existence and uniqueness of the solution $(\mathbf{u},\mathbf{H},p)$ of the problem (69), (70) (for given $T\in H^1\left(\mathsf{\Omega }\right)$).

The above features allow us to easily prove the solvability of the general linear MHDB problem (69)–(71), to derive the corresponding estimates of the solution and to establish the isomorphism of the linear operator corresponding to the above problem. The latter will play an important role in the study of control problems and, in particular, when deriving the optimality system. To prove the above facts, we introduce linear operators

$$A:H\equiv {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\to H_{0T}^{*},B:H_{0T}\to L_0^2{\left(\mathsf{\Omega }\right)}^{*}\equiv L_0^2\left(\mathsf{\Omega }\right),$$

$$B^{*}:L_0^2\left(\mathsf{\Omega }\right)\to H_{0T}^{*},B_1:H^1\left(\mathsf{\Omega }\right)\to H_{0T}^{*},A_2:H^1\left(\mathsf{\Omega }\right)\to {\mathcal{T}}^{*},$$

which are defined by the formulas

$$\langle A(\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi })\rangle =a((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))+(({\mathbf{u}}^0·\nabla )\mathbf{u},\mathbf{v})+$$

$$\kappa [(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}^0,\mathbf{u})-(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}^0,\mathbf{v})]\forall (\mathbf{u},\mathbf{H})\in H,(\mathbf{v},\mathbf{\Psi })\in H_{0T},$$

$$\langle A_{2}T,S\rangle =a_2(T,S)+({\mathbf{u}}^0·\nabla T,S)\forall T\in H^1\left(\mathsf{\Omega }\right),S\in \mathcal{T},$$

$$\langle B(\mathbf{v},\mathbf{\Psi }),r\rangle =b(\mathbf{v},r)=\langle B^{*}r,(\mathbf{v},\mathbf{\Psi })\rangle \forall (\mathbf{v},\mathbf{\Psi })\in H_{0T},r\in L_0^2\left(\mathsf{\Omega }\right),$$

$$\langle B_{1}T,\left(\mathbf{v},\mathbf{\Psi }\right)\rangle =b_1(T,\mathbf{v})\equiv (\mathbf{b}T,\mathbf{v})\forall T\in H^1\left(\mathsf{\Omega }\right),(\mathbf{v},\mathbf{\Psi })\in H_{0T}.$$

It follows from the properties of the bilinear and trilinear forms underlying the defined operators that the operators A, B, $B^{*}$, $B_1$ and $A_2$ are continuous, and the operators A and $A_2$ are, moreover, coercive on the spaces $V_{0T}$ and $\mathcal{T}$, respectively. Reasoning further, as in Alekseev [13], we easily show that the problem (69)–(71) has a unique solution $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$, that depends continuously on the initial data. This forms the content of the following lemma.

Lemma 5.

When Hypothesises 1, 2 and 6 are satisfied, the problem (69)–(71) for any septenary $(\mathbf{F},\sigma ,\mathbf{g},q,l,\psi ,\alpha )\in H_{0T}^{*}\times L_0^2\left(\mathsf{\Omega }\right)\times {\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)\times {\tilde{H}}^s\left(\mathsf{\Gamma }\right)\times {\mathcal{T}}^{*}\times H^{1/2}\left({\mathsf{\Gamma }}_D\right)\times L_{+}^2\left({\mathsf{\Gamma }}_N\right)$ has a unique solution $(\mathbf{u},\mathbf{H},T,p)\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right)$ where $r=s+1/2$, and the following estimates are valid:

$${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\le {\widehat{M}}_{\mathbf{u}}{,\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le {\widehat{M}}_{\mathbf{H}}{,\parallel T\parallel }_{1,\mathsf{\Omega }}\le {\widehat{M}}_T,{\parallel p\parallel }_{\mathsf{\Omega }}\le {\widehat{M}}_p.$$

Here, ${\widehat{M}}_{\mathbf{u}}$, ${\widehat{M}}_{\mathbf{H}}$, ${\widehat{M}}_T$ and ${\widehat{M}}_p$ are continuous non-decreasing functions of norms ${\parallel \mathbf{F}\parallel }_{H_{0T}^{*}}$, ${\parallel \sigma \parallel }_{\mathsf{\Omega }}$, ${\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}$, ${\parallel q\parallel }_{s,\mathsf{\Gamma }}$, ${\parallel l\parallel }_{{\mathcal{T}}^{*}}$, ${\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}$ and ${\parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}$.

Supposing that

$$X={\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\times H^1\left(\mathsf{\Omega }\right)\times L_0^2\left(\mathsf{\Omega }\right),\mathbf{x}=(\mathbf{u},\mathbf{H},T,p),$$

$$Y=H_{0T}^{*}\times L_0^2\left(\mathsf{\Omega }\right)\times {\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)\times {\tilde{H}}^s\left(\mathsf{\Gamma }\right)\times {\mathcal{T}}^{*}\times H^{1/2}\left({\mathsf{\Gamma }}_D\right),$$

(73)

let us introduce the operator $\Phi =({\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4,{\Phi }_5,{\Phi }_6):X\to Y$, where

$${\Phi }_1\left(\mathbf{x}\right)=A\left((\mathbf{u},\mathbf{H})\right)+B^{*}p+B_{1}T,{\Phi }_2\left(\mathbf{x}\right)\equiv \mathrm{div}\mathbf{u},$$

$${\Phi }_3{\left(\mathbf{x}\right)=\mathbf{u}|}_{\mathsf{\Gamma }},{\Phi }_4\left(\mathbf{x}\right)=\mathbf{H}·\mathbf{n}{{|}_{\mathsf{\Gamma }},{\Phi }_5\left(\mathbf{x}\right)=A_{2}T,{\Phi }_6\left(\mathbf{x}\right)=T|}_{{\mathsf{\Gamma }}_D}.$$

(74)

By virtue of Lemma 5 the following result holds.

Theorem 2.

If Hypothesises 1, 2 and 6 are satisfied, the operator $\Phi :X\to Y$ defined in (73),(74), realizes the isomorphism of the spaces X and Y.

4. Formulation and Analysis of Control Problems

Let us proceed to the formulation of control problems for the MHD model under consideration. For this purpose, first of all, it is necessary to decide on the choice of possible controls, i.e., those functions included in the set ($\mathbf{f}$, $\mathbf{j}$, f, $\mathbf{b}$, $\mathbf{g}$, q, $\mathbf{k}$, $\psi$, $\chi$, $\alpha$) of the initial data of Problem 1 that can be changed in the process of solving the control problem in order to achieve its goals. In accordance with the MHD model used by us, which takes into account three main physical processes: hydrodynamic, thermal, and electromagnetic, it is convenient to categorize all possible controls into three classes: hydrodynamic, thermal and electromagnetic.

It is usual to choose the boundary vector $\mathbf{g}$ as the hydrodynamic type control. As a temperature type control, we will choose two boundary functions $\psi$ and $\chi$ included in the boundary conditions for the temperature T in (5). Finally, we choose as electromagnetic type controls the function q included in the boundary condition for the magnetic field $\mathbf{H}$ in (4). About the physical meaning of choosing the above functions as controls, see, e.g., Alekseev ([14], Sections 2, 3 and 6).

We will assume that the group of fixed data consists of $\mathbf{f}$, $\mathbf{b}$, f, $\mathbf{k}$ and $\alpha$, while the group of controls consists of $\mathbf{g},q,\psi$ and $\chi$, which can change in some sets $K_1$, $K_2$, $K_3$ and $K_4$. Besides, the function (exterior current density) $\mathbf{j}$ will play a special role in the sense that below we will study the stability of solutions of the control problems under consideration with respect to small perturbations of both the cost functional under minimization and of the function $\mathbf{j}$ (in the norm of ${\mathbf{L}}^2\left(\mathsf{\Omega }\right)$).

More precisely, we denote by $I:X\to \mathbb{R}$ the weakly semi-continuous from below functional, by ${\mu }_l$—the non-negative constants, $l=0,1,2,3,4$, and assume in addition to Hypothesises 1–4 that the following take place:

Hypothesis 8.

$K_1\subset {\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right)$, $K_2\subset {\tilde{H}}^s\left(\mathsf{\Gamma }\right)$, $K_3\subset H^{1/2}\left({\mathsf{\Gamma }}_D\right)$, $K_4\subset L^2\left({\mathsf{\Gamma }}_N\right)$ are non-empty closed convex sets, $0<s\le 1/2$;

Hypothesis 9.

${\mu }_0=\mathrm{const}>0$, ${\mu }_l=\mathrm{const}\ge 0$ and $K_l$ are bounded sets or ${\mu }_l>0$ and the functional I is bounded from below, $1\le l\le 4$.

Let us put $K\equiv K_1\times K_2\times K_3\times K_4$, $\mathbf{x}\equiv (\mathbf{u},\mathbf{H},T,p)$, $u_0=(\mathbf{f},\mathbf{b},f,\mathbf{k},\alpha )$, $u\equiv (\mathbf{g},q,\psi ,\chi )$ and define the functional $J:X\times K\to \mathbb{R}$ under minimization by the formula

$$J(\mathbf{x},u)=\frac{{\mu }_0}{2}I\left(\mathbf{x}\right)+\frac{{\mu }_1}{2}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\frac{{\mu }_2}{2}{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\frac{{\mu }_3}{2}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\frac{{\mu }_4}{2}{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2.$$

(75)

Let us write the weak formulation (54)–(56) of Problem 1 as

$$F(\mathbf{x},u,\mathbf{j})\equiv F(\mathbf{u},\mathbf{H},T,p,\mathbf{g},q,\psi ,\chi ,\mathbf{j})=0.$$

(76)

Here, $F\equiv (F_1,F_2,F_3,F_4,F_5,F_6):X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right)\to Y$ describes the operator acting according to

$$\langle F_1(\mathbf{x},u,\mathbf{j}),(\mathbf{v},\mathbf{\Psi })\rangle =a((\mathbf{u},\mathbf{H}),(\mathbf{v},\mathbf{\Psi }))+((\mathbf{u}·\nabla )\mathbf{u},\mathbf{v})+b(\mathbf{v},p)+$$

$$+b_1(T,\mathbf{v})+\kappa [(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},\mathbf{u})-(\mathrm{rot}\mathbf{H}\times \mathbf{H},\mathbf{v})]-\langle \mathbf{F},(\mathbf{v},\mathbf{\Psi })\rangle -$$

$$-\langle f,\mathbf{v}\rangle -{\nu }_1(\mathbf{j},\mathrm{rot}\mathbf{\Psi })-{(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}.$$

$$F_2(\mathbf{x},u)=\mathrm{div}\mathbf{u},F_3{(\mathbf{x},u)=\mathbf{u}|}_{\mathsf{\Gamma }}-\mathbf{g},F_4(\mathbf{x},u)=\mathbf{H}·\mathbf{n}-q,$$

$$\langle F_5(\mathbf{x},u),S\rangle =\lambda \tilde{a}(T,S)+\lambda {(\alpha T,S)}_{{\mathsf{\Gamma }}_N}+(\mathbf{u}·\nabla T,S)-\langle f,S\rangle -{(\chi ,S)}_{{\mathsf{\Gamma }}_N},$$

$$F_6{(\mathbf{x},u)=T|}_{{\mathsf{\Gamma }}_D}-\psi .$$

(77)

When writing the boundary value problem (54)–(56) in the equivalent form (76), (77), we considered that the system (1)–(5) contains eleven relations that are used in the formation of all six components $F_1$, $F_2$, $F_3$, $F_4$, $F_5$ and $F_6$ of the operator F, as given in (77).

Let us set the following control problem:

$$J(\mathbf{x},u)\equiv \frac{{\mu }_0}{2}I\left(\mathbf{x}\right)+\frac{{\mu }_1}{2}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\frac{{\mu }_2}{2}{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\frac{{\mu }_3}{2}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\frac{{\mu }_4}{2}{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\to inf,$$

$$F(\mathbf{x},u,\mathbf{j})=0,(\mathbf{x},u)\in X\times K.$$

(78)

We will consider the following functionals as cost functionals:

$$I_1\left(\mathbf{x}\right)=\parallel \mathbf{u}-{\mathbf{v}}_d{\parallel }_Q^2,I_2\left(\mathbf{x}\right)=\parallel \mathbf{H}-{\mathbf{H}}_d{\parallel }_Q^2,I_3\left(T\right)={\parallel T-T_d\parallel }_Q^2,$$

$$I_4\left(\mathbf{x}\right)=\parallel \mathrm{rot}\mathbf{u}-{\eta }_d{\parallel }_Q^2,I_5\left(\mathbf{x}\right)={\parallel p-p_d\parallel }_Q^2,$$

$$I_6\left(\mathbf{x}\right)={|\mathbf{u}|}_{1,\mathsf{\Omega }}^2,{\tilde{I}}_7\left(\mathbf{x}\right)={\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2.$$

(79)

Here, ${\mathbf{v}}_d\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}_d\in {\mathbf{L}}^2\left(Q\right)$, $T_d\in L^2\left(Q\right)$, ${\eta }_d\in {\mathbf{L}}^2\left(Q\right)$, $p_d\in L^2\left(Q\right)$ are given functions.

By $Z_{ad}=\{(\mathbf{x},u)\in X\times K:F(\mathbf{x},u)=0,J(\mathbf{x},u)<\infty \}$ we denote the set of admissible pairs for the problem (78).

Theorem 3.

Suppose that Hypothesises 1–4, 8 and 9 are satisfied, the functional $I:X\to \mathbb{R}$ is weakly semi-continuous from below on X and $Z_{ad}\ne \varnothing$. Then there exists at least one solution to the control problem (78).

Proof of Theorem 3.

We denote by $({\mathbf{x}}_m$, $u_m)\equiv ({\mathbf{u}}_m$, ${\mathbf{H}}_m$, $T_m$, $p_m$, ${\mathbf{g}}_m$, $q_m$, ${\psi }_m$, ${\chi }_m)\in Z_{ad}$ the minimizing sequence for which

$$\underset{m\to \infty }{lim}J({\mathbf{x}}_m,u_m)=\underset{(\mathbf{x},u)\in Z_{ad}}{inf}J(\mathbf{x},u)\equiv J^{*}.$$

By virtue of Hypothesises 8 and 9 and Theorem 1, the next estimates for the controls $u_m=({\mathbf{g}}_m,q_m,{\psi }_m,{\chi }_m)$ and for corresponding solutions ${\mathbf{x}}_m=({\mathbf{u}}_m$, ${\mathbf{H}}_m$, $T_m$, $p_m)$ of Problem 1 are fulfilled

$$\parallel {\mathbf{g}}_m{\parallel }_{1/2,\mathsf{\Gamma }}\le c_1,\parallel q_m{\parallel }_{s,\mathsf{\Gamma }}\le c_2,\parallel {\psi }_m{\parallel }_{1/2,{\mathsf{\Gamma }}_D}\le c_3,{\parallel {\chi }_m\parallel }_{{\mathsf{\Gamma }}_N}\le c_4,$$

(80)

$$\parallel {\mathbf{u}}_m{\parallel }_{1,\mathsf{\Omega }}\le c_5,\parallel {\mathbf{H}}_m{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le c_6\parallel T_m{\parallel }_{1,\mathsf{\Omega }}\le c_7,{\parallel p_m\parallel }_{\mathsf{\Omega }}\le c_8.$$

(81)

Here, and below $c_1,c_2$, … are constants independent of m. From (80), (81) it follows that there exist weak limits ${\mathbf{g}}^{*}\in K_1$, $q^{*}\in K_2$, ${\psi }^{*}\in K_3$, ${\chi }^{*}\in K_4$, ${\mathbf{u}}^{*}\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)$, ${\mathbf{H}}^{*}\in {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$, $T^{*}\in H^1\left(\mathsf{\Omega }\right)$, $p^{*}\in L_0^2\left(\mathsf{\Omega }\right)$ of some subsequences of sequences $\left\{{\mathbf{g}}_m\right\}$, $\left\{q_m\right\}$, $\left\{{\psi }_m\right\}$, $\left\{{\chi }_m\right\}$, $\left\{{\mathbf{u}}_m\right\}$, $\left\{{\mathbf{H}}_m\right\}$, $\left\{T_m\right\}$, $\left\{p_m\right\}$. Taking this into account, we can suppose that for $m\to \infty$ the following limit transitions are valid:

$${\mathbf{g}}_m\to {\mathbf{g}}^{*}\mathrm{weakly}\mathrm{in}{\mathbf{H}}_T^{1/2}\left(\mathsf{\Gamma }\right),q_m\to q^{*}\mathrm{weakly}\mathrm{in}{\tilde{H}}^s\left(\mathsf{\Gamma }\right),$$

$${\psi }_m\to {\psi }^{*}\mathrm{weakly}\mathrm{in}{H}^{1/2}\left({\mathsf{\Gamma }}_D\right),{\chi }_m\to {\chi }^{*}\mathrm{weakly}\mathrm{in}{L}^2\left({\mathsf{\Gamma }}_N\right),$$

$${\mathbf{u}}_m\to {\mathbf{u}}^{*}\mathrm{weakly}\mathrm{in}{\mathbf{H}}_1\left(\mathsf{\Omega }\right)\mathrm{and}\mathrm{strongly}\mathrm{in}{\mathbf{L}}^4\left(\mathsf{\Omega }\right),$$

$${\mathbf{H}}_m\to {\mathbf{H}}^{*}\mathrm{weakly}\mathrm{in}{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\cap {\mathbf{L}}^3\left(\mathsf{\Omega }\right)\mathrm{and}\mathrm{strongly}\mathrm{in}{\mathbf{L}}^2\left(\mathsf{\Omega }\right),$$

$$T_m\to T^{*}\mathrm{weakly}\mathrm{in}{H}^1\left(\mathsf{\Omega }\right)\mathrm{and}\mathrm{strongly}\mathrm{in}{L}^4\left(\mathsf{\Omega }\right),$$

$$p_m\to p^{*}\mathrm{weakly}\mathrm{in}{L}^2\left(\mathsf{\Omega }\right).$$

(82)

Since ${\mathbf{u}}_m{|}_{\mathsf{\Gamma }}\equiv \gamma {\mathbf{u}}_m={\mathbf{g}}_m$, then from the continuity of the trace operator $\gamma :{\mathbf{H}}^1\left(\mathsf{\Omega }\right)\to {\mathbf{H}}^{1/2}\left(\mathsf{\Gamma }\right)$ it follows that ${\mathbf{u}}^{*}{|}_{\mathsf{\Gamma }}={\mathbf{g}}^{*}$, so that $F_3({\mathbf{x}}^{*},u^{*})=0$, where ${\mathbf{x}}^{*}=({\mathbf{u}}^{*},{\mathbf{H}}^{*},T^{*},p^{*})$, $u^{*}=({\mathbf{g}}^{*},q^{*},{\psi }^{*},{\chi }^{*})$.

Using the continuity of the operator $\mathrm{div}$ in ${\mathbf{H}}^1\left(\mathsf{\Omega }\right)$ and the continuity of the trace operators ${\gamma }_n:{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)\to H^s\left(\mathsf{\Gamma }\right)$ at $s=r-1/2$ and ${\gamma |}_{{\mathsf{\Gamma }}_D}:H^1\left(\mathsf{\Omega }\right)\to H^{1/2}\left({\mathsf{\Gamma }}_D\right)$, we similarly show that

$$F_3({\mathbf{x}}^{*},u^{*})=0,F_4({\mathbf{x}}^{*},u^{*})=0,F_6({\mathbf{x}}^{*},u^{*})=0.$$

(83)

It remains to prove that the elements $({\mathbf{u}}^{*},{\mathbf{H}}^{*},T^{*},p^{*})$ and $({\mathbf{g}}^{*},q^{*},{\psi }^{*},{\chi }^{*})$ satisfy the relations $F_1({\mathbf{x}}^{*},u^{*},\mathbf{j})=0$ and $F_5({\mathbf{x}}^{*},u^{*})=0$, which by virtue of (77) are equivalent to the identities

$$\nu (\nabla {\mathbf{u}}^{*},\nabla \mathbf{v})+{\nu }_1(\mathrm{rot}{\mathbf{H}}^{*},\mathrm{rot}\mathbf{\Psi })+(({\mathbf{u}}^{*}·\nabla ){\mathbf{u}}^{*},\mathbf{v})-$$

$$-(\mathrm{div}\mathbf{v},p^{*})+\kappa [(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}^{*},{\mathbf{u}}^{*})-(\mathrm{rot}{\mathbf{H}}^{*}\times {\mathbf{H}}^{*},\mathbf{v})]+b(\mathbf{v},p^{*})+b_1(T^{*},\mathbf{v})=$$

$$=\langle \mathbf{f},\mathbf{v}\rangle +(\mathbf{j},\mathrm{rot}\mathbf{\Psi })+{(\mathbf{k},{\mathbf{\Psi }}_T)}_{\mathsf{\Gamma }}\forall (\mathbf{v},\mathbf{\Psi })\in H_{0T},$$

(84)

and

$$\lambda \tilde{a}(T^{*},S)+\lambda {({\alpha }^{*}{T}^{*},S)}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}^{*}·\nabla T^{*},S)=\langle f,S\rangle +{({\chi }^{*},S)}_{{\mathsf{\Gamma }}_N}\forall S\in \mathcal{T}.$$

(85)

For this purpose, we note that for each $m=1,2,\dots$ the quadruple (${\mathbf{u}}_m$, ${\mathbf{H}}_m$, $T_m$, $p_m$) satisfies the relations $F_1({\mathbf{x}}_m$, $u_m,\mathbf{j})=0$ and $F_5({\mathbf{x}}_m$, $u_m)=0$ equivalent to identities:

$$\nu (\nabla {\mathbf{u}}_m,\nabla \mathbf{v})+{\nu }_1(\mathrm{rot}{\mathbf{H}}_m,\mathrm{rot}\mathbf{\Psi })+(({\mathbf{u}}_m·\nabla ){\mathbf{u}}_m,\mathbf{v})-$$

$$-(\mathrm{div}\mathbf{v},p_m)+\kappa [(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}_m,{\mathbf{u}}_m)-(\mathrm{rot}{\mathbf{H}}_m\times {\mathbf{H}}_m,\mathbf{v})]+b(\mathbf{v},p_m)+b_1(T_m,\mathbf{v})=$$

$$=\langle \mathbf{f},\mathbf{v}\rangle +(\mathbf{j},\mathrm{rot}\mathbf{\Psi })+{(\mathbf{k},\mathbf{\Psi })}_{\mathsf{\Gamma }}\forall (\mathbf{v},\mathbf{\Psi })\in H_{0T},$$

(86)

and

$$\lambda \tilde{a}(T_m,S)+\lambda {(\alpha T_m,S)}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_m·\nabla T_m,S)=\langle f,S\rangle +{({\chi }_m,S)}_{{\mathsf{\Gamma }}_N}\forall S\in \mathcal{T}.$$

(87)

It follows from (82) that all linear summands in identities (86) and (87) pass as $m\to \infty$ to the corresponding linear summands in (84) and (85).

Arguing as in Alekseev ([14], Section 4), or as in Alekseev [58], we show that the nonlinear hydrothermodynamic summands $(({\mathbf{u}}_m·\nabla ){\mathbf{u}}_m,\mathbf{v})$ and $({\mathbf{u}}_m·\nabla T_m,S)$ tend, respectively, to $(({\mathbf{u}}^{*}·\nabla ){\mathbf{u}}^{*},\mathbf{v})$ and $({\mathbf{u}}^{*}·\nabla T^{*},S)$ as $m\to \infty$, whereas the nonlinear magnetohydrodynamic summand $(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}_m,{\mathbf{u}}_m)-(\mathrm{rot}{\mathbf{H}}_m\times {\mathbf{H}}_m,\mathbf{v})$ tends to $(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}^{*},{\mathbf{u}}^{*})-(\mathrm{rot}{\mathbf{H}}^{*}\times {\mathbf{H}}^{*}\times {\mathbf{H}}^{*},\mathbf{v})$ when $m\to \infty$. This means that $F_1({\mathbf{x}}^{*}$, $u^{*},\mathbf{j})=0$, $F_5({\mathbf{x}}^{*}$, $u^{*})=0$. Finally, considering the weak semi-continuity from below of the functional I on X, we conclude that $J({\mathbf{x}}^{*}$, $u^{*})=J^{*}$. □

It should be noted that each of the functionals $I_1,I_2,\dots ,I_7$ defined in (79) is weakly semi-continuous from below. Taking into account these facts and Theorem 3, we have the following corollary.

Corollary 1.

Suppose that Hypothesises 1–4 and 8 are satisfied, ${\mu }_0>0$ and, in addition, ${\mu }_l>0$ or ${\mu }_l\ge 0$ and $K_l$ are bounded sets at $1\le l\le 4$. Then there exists at least one solution to the problem (78) for $I=I_k$, $1\le k\le 7$.

The next stage of our study consists in justifying the application of the principle of Lagrange indefinite multipliers to the problem (75). For this purpose, we use the extremum principle in smooth-convex conditional minimization problems (see the books by Ioffe et al. and Fursikov, respectively, refs. [74,75], and also the book by Alekseev ([14], Appendix 1)).

Let us first introduce dual spaces

$$X^{*}={\mathbf{H}}_T^{-1}\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r{\left(\mathsf{\Omega }\right)}^{*}\times H^1{\left(\mathsf{\Omega }\right)}^{*}\times L_0^2\left(\mathsf{\Omega }\right)$$

and

$$Y^{*}\equiv H_{0T}\times L_0^2\left(\mathsf{\Omega }\right)\times {\mathbf{H}}_T^{-1/2}\left(\mathsf{\Gamma }\right)\times {\tilde{H}}^{-s}\left(\mathsf{\Gamma }\right)\times \mathcal{T}\times H^{1/2}{\left({\mathsf{\Gamma }}_D\right)}^{*}$$

of the products X and Y defined in (73). Let us denote by $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}):X\to Y$ the Fréchet derivative of F on the state $\mathbf{x}$ at a point $(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})$. By $F_{\mathbf{x}}^{\prime }{(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})}^{*}:Y^{*}\to X^{*}$ we denote the conjugate of $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})$ an operator uniquely defined by $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})$ by the relation

$${\langle F_{\mathbf{x}}^{\prime }{(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})}^{*}{\mathbf{y}}^{*},\mathbf{x}\rangle }_{X^{*}\times X}={\langle {\mathbf{y}}^{*},F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})\mathbf{x}\rangle }_{Y^{*}\times Y}\forall \mathbf{x}\in X,{\mathbf{y}}^{*}\in Y^{*}.$$

In accordance with the general theory of extremum problems (see the book by Ioffe [74]), let us introduce the element ${\mathbf{y}}^{*}=((\xi ,\eta ),\sigma ,{\zeta }_1,{\zeta }_2,\theta ,{\zeta }_3)\in Y^{*}$, which we will refer to below as an adjoint state, and the Lagrangian $\mathcal{L}:X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right)\times {\mathbb{R}}^{+}\times Y^{*}\to \mathbb{R}$, where ${\mathbb{R}}^{+}=\{x\in \mathbb{R}:x\ge 0\}$, by formula

$$\mathcal{L}(\mathbf{x},u,\mathbf{j},{\lambda }_0,{\mathbf{y}}^{*})={\lambda }_{0}J(\mathbf{x},u)+{\langle {\mathbf{y}}^{*},F(\mathbf{x},u,\mathbf{j})\rangle }_{Y^{*}\times Y}\equiv {\lambda }_{0}J(\mathbf{x},u)+$$

$${\langle F_1(\mathbf{x},u,\mathbf{j}),(\xi ,\eta )\rangle }_{H_{0T}^{*}\times H_{0T}}+(F_2(\mathbf{x},u),\sigma )+{\langle {\zeta }_1,F_3(\mathbf{x},u)\rangle }_{\mathsf{\Gamma }}+{\langle {\zeta }_2,F_4(\mathbf{x},u)\rangle }_{\mathsf{\Gamma }}+$$

$$+\tilde{\kappa }\langle F_5(\mathbf{x},u),\theta \rangle +\tilde{\kappa }{\langle {\zeta }_3,F_6(\mathbf{x},u)\rangle }_{{\mathsf{\Gamma }}_D}.$$

(88)

In (88), ${\lambda }_0$ is a non-negative dimensionless multiplier, while $\tilde{\kappa }$ is a dimensional parameter whose value is 1 so that the following relations for the dimensions $\left[{\lambda }_0\right]$ and $\left[\tilde{\kappa }\right]$ are satisfied

$$\left[{\lambda }_0\right]=1,\left[\tilde{\kappa }\right]=L_0^2{T}_0^{-2}{K}_0^{-2}.$$

(89)

As in the case of Formula (36) in Section 2, the sense of introduction of the parameter $\tilde{\kappa }$ into (88) is to equalize the dimensions of all summands in the right-hand side of (88), as well as to synchronize below the dimensions of magnetohydrodynamic and temperature variables of both main and adjoint states. It is easy to check, in particular, that all summands of the identity obtained after multiplication by $\tilde{\kappa }$ of identity (55) for T have the same dimension as all summands in the magnetohydrodynamic identity (54).

From the Formula (88) it follows that if conditions (89) are satisfied, the dimensions of all Lagrangian multipliers $(\xi ,\eta ),\sigma ,{\zeta }_1,{\zeta }_2,\theta ,{\zeta }_3$ are determined by the dimension of the value $J(\mathbf{x},u)$ of functional J, i.e., in essence, by the dimension of parameter ${\mu }_0$ in the expression (75). Below we will always choose the dimension of $\left[{\mu }_0\right]$ such that the following relations hold:

$$\left[\xi \right]=\left[\mathbf{u}\right]=L_0{T}_0^{-1},\left[\eta \right]=\left[\mathbf{H}\right]=I_0{L}_0^{-1},$$

$$\left[\theta \right]=\left[T\right]=K_0,\left[\sigma \right]=\left[p\right]=L_0^2{T}_0^{-2}.$$

(90)

Accounting for (90), we will refer below to the components $\xi ,\theta ,\eta$, and $\sigma$ as adjoint velocity, temperature, magnetic field and pressure. The following theorem gives sufficient conditions on the initial problem data to ensure the Fredholm property of the operator $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}):X\to Y$.

Theorem 4.

When Hypothesises 1–4, 8 and 9 are satisfied, the operator $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}):X\to Y$ is Fredholm for any pair $(\widehat{\mathbf{x}},\widehat{u})\in X\times K$.

Proof of Theorem 4.

To prove the Fredholm property, it is enough to calculate the Fréchet derivative at $\mathbf{x}$ of the operator $F:X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right)\to Y$ defined in (77). Standard reasoning shows that at any point $(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})\equiv (\widehat{\mathbf{u}},\widehat{\mathbf{H}},\widehat{T},\widehat{p},\widehat{\mathbf{g}},\widehat{q},\widehat{\psi },\widehat{\chi },\mathbf{j})\in X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$ the mentioned Frechet derivative is a linear continuous operator $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}):X\to Y$, putting in correspondence to each element $(\mathbf{w},\mathbf{h},r,\tau )\in X$ the element $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})$ $(\mathbf{w},\mathbf{h},r,\tau )=(\widehat{y_1},\widehat{y_2},\widehat{y_3},\widehat{y_4},\widehat{y_5},\widehat{y_6})\in Y$, where

$$\langle {\widehat{y}}_1,(\mathbf{v},\mathbf{\Psi })\rangle =\nu a_0(\mathbf{w},\mathbf{v})+{\nu }_1{a}_1(\mathbf{h},\mathbf{\Psi })+[((\widehat{\mathbf{u}}·\nabla )\mathbf{w},\mathbf{v})+((\mathbf{w}·\nabla )\widehat{\mathbf{u}},\mathbf{v})]+$$

$$+b(\mathbf{v},r)+b_1(\tau ,\mathbf{v})-\kappa [(\mathrm{rot}\widehat{\mathbf{H}}\times \mathbf{h},\mathbf{v})+(\mathrm{rot}\mathbf{h}\times \widehat{\mathbf{H}},\mathbf{v})]+\kappa [(\mathrm{rot}\mathbf{\Psi }\times \mathbf{h},\widehat{\mathbf{u}})+$$

$$+(\mathrm{rot}\mathbf{\Psi }\times \widehat{\mathbf{H}},\mathbf{w})]\forall \left(\mathbf{v},\mathbf{\Psi }\right)\in H_{0T},\langle {\widehat{y}}_2,r\rangle =b(\mathbf{w},r)\forall r\in L_0^2\left(\mathsf{\Omega }\right),$$

$${\widehat{y}}_3=\mathbf{w}{{|}_{\mathsf{\Gamma }},{\widehat{y}}_4=\mathbf{h}·\mathbf{n}|}_{\mathsf{\Gamma }},\langle {\widehat{y}}_5,S\rangle =\lambda \tilde{a}(\tau ,S)+\lambda {(\widehat{\alpha }\tau ,S)}_{{\mathsf{\Gamma }}_N}+$$

$$+(\widehat{\mathbf{u}}·\nabla \tau ,S)+(\mathbf{w}·\nabla \widehat{T},S)\forall S\in \mathcal{T},{\widehat{y}}_6{=\tau |}_{{\mathsf{\Gamma }}_D}.$$

(91)

From (91) we conclude that

$$F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u})=\Phi +\widehat{\Phi }\equiv ({\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4,{\Phi }_5,{\Phi }_6)+({\widehat{\Phi }}_1,0,0,0,{\widehat{\Phi }}_5,0).$$

Here, the operators ${\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4,{\Phi }_5$ and ${\Phi }_6$ were introduced in (74), and the operators ${\widehat{\Phi }}_1:X\to H_{0T}^{*}$ and ${\widehat{\Phi }}_5:X\to {\mathcal{T}}^{*}$ are defined by the formulas

$$\langle {\widehat{\Phi }}_1(\mathbf{w},\mathbf{h},r,\tau ),(\mathbf{v},\mathbf{\Psi })\rangle =((\mathbf{w}·\nabla )\widehat{\mathbf{u}},\mathbf{v})+\kappa [(\mathrm{rot}\mathbf{\Psi }\times \mathbf{h},\widehat{\mathbf{u}}-(\mathrm{rot}\widehat{\mathbf{H}}\times \mathbf{h},\mathbf{v})],$$

$$\langle {\widehat{\Phi }}_5(\mathbf{w},\mathbf{h},r,\tau ),S\rangle =\tilde{\kappa }(\mathbf{w}·\nabla \widehat{T},S).$$

By virtue of Theorem 2, the operator $\Phi :X\to Y$ is an isomorphism, and from (24)–(27) and (41) it follows that the operator $({\widehat{\Phi }}_1,{\widehat{\Phi }}_5)$ is continuous from ${\mathbf{L}}^4\left(\mathsf{\Omega }\right)\times {\mathbf{L}}^3\left(\mathsf{\Omega }\right)\times L^4\left(\mathsf{\Omega }\right)$ to ${\mathbf{H}}^{-1}\left(\mathsf{\Omega }\right)\times {\mathbf{V}}_T^{*}\times {\mathcal{T}}^{*}$, and hence it is compact from ${\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$ in ${\mathbf{H}}^{-1}\left(\mathsf{\Omega }\right)\times {\mathbf{V}}_T^{*}\times {\mathcal{T}}^{*}$. This is equivalent to the Fredholm property of $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u})$. □

The following theorem provides a justification of the validity of the Lagrange principle for the control problem (78).

Theorem 5.

Suppose that if Hypothesises 1–4, 8 and 9 hold, the element $(\widehat{\mathbf{x}},\widehat{u})\equiv (\widehat{\mathbf{u}},\widehat{\mathbf{H}},\widehat{T},\widehat{p},\widehat{\mathbf{g}},\widehat{q},\widehat{\psi },\widehat{\chi })$ $\in X\times K$ is the point of local minimum for problem (78), and the functional I is continuously differentiable with respect to $\mathbf{x}$ at the point $(\widehat{\mathbf{x}},\widehat{u})$ for any element $u\in K$ and convex with respect to u for every point $\mathbf{x}\in X$. Then there exists a nonzero Lagrange multiplier $({\lambda }_0,{\mathbf{y}}^{*})\in {\mathbb{R}}^{+}\times Y^{*}$ for which the Euler–Lagrange equation has the form

$$F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}){\mathbf{y}}^{*}+{\lambda }_0({\mu }_0/2)I_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u})=0in{X}^{*}$$

(92)

is valid that is equivalent to identities

$$\nu a_0(\mathbf{w},\xi )+{\nu }_1{a}_1(\mathbf{h},\eta )+((\widehat{\mathbf{u}}·\nabla )\mathbf{w},\xi )+$$

$$+((\mathbf{w}·\nabla )\widehat{\mathbf{u}},\xi )+\kappa [(\mathrm{rot}\eta \times \widehat{\mathbf{H}},\mathbf{w})+(\mathrm{rot}\eta \times \mathbf{h},\widehat{\mathbf{u}})]$$

$$-\kappa [(\mathrm{rot}\widehat{\mathbf{H}}\times \mathbf{h},\xi )+(\mathrm{rot}\mathbf{h}\times \widehat{\mathbf{H}},\xi )]+\tilde{\kappa }(\mathbf{w}·\nabla \widehat{T},\theta )+$$

$$+b(\mathbf{w},\sigma )+{\langle {\zeta }_1,\mathbf{w}\rangle }_{\mathsf{\Gamma }}+{\langle {\zeta }_2,\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}+{\lambda }_0({\mu }_0/2)\langle I_{\mathbf{u}}^{\prime }\left(\widehat{\mathbf{x}}\right),\mathbf{w}\rangle +$$

$$+{\lambda }_0({\mu }_0/2)\langle I_{\mathbf{H}}^{\prime }\left(\widehat{\mathbf{x}}\right),\mathbf{h}\rangle =0\forall (\mathbf{w},\mathbf{h})\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right),$$

(93)

$$\tilde{\kappa }[\lambda \tilde{a}(\tau ,\theta )+\lambda {(\widehat{\alpha }\tau ,\theta )}_{{\mathsf{\Gamma }}_N}+(\widehat{\mathbf{u}}·\nabla \tau ,\theta )]+$$

$$+{\langle {\zeta }_3,\tau \rangle }_{{\mathsf{\Gamma }}_D}+{\lambda }_0({\mu }_0/2)\langle I_T^{\prime }\left(\widehat{\mathbf{x}}\right),\tau \rangle +b_1(\tau ,\xi )=0\forall \tau \in \mathcal{T},$$

(94)

$$b(\xi ,r)+{\lambda }_0(J_p^{\prime }\left(\widehat{\mathbf{x}}\right),r)=0\forall r\in L_0^2\left(\mathsf{\Omega }\right),$$

(95)

and the minimum principle is satisfied, which has the form of inequality

$$\mathcal{L}(\widehat{\mathbf{x}},\widehat{u},{\lambda }_0,{\mathbf{y}}^{*})\le \mathcal{L}(\widehat{\mathbf{x}},u,{\lambda }_0,{\mathbf{y}}^{*})\forall u\in K.$$

(96)

Proof of Theorem 5

Let us use the extremum principle in smooth-convex conditional minimization problems (see Ioffe et al. and Fursikov, respectively, refs. [74,75], and also Alekseev ([14], Appendix 1)). It is easy to see that the set $F(\widehat{\mathbf{x}},K)$ is a convex set in Y. Therefore, the statement of Theorem 5 follows from the Fredholm property of the operator $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j}):X\to Y$. □

Let us turn to the inequalities (62), (63), which ensure the uniqueness of the solution of Problem 1. It is clear that for fixed elements $\mathbf{f}$, $\mathbf{b}$, f, $\mathbf{k}$ and $\alpha$, the right-hand sides $M_{\mathbf{u}}$, $M_{\mathbf{H}}$, $M_T$ and $M_p$ included in the inequalities (62), (63) depend on the control $u=(\mathbf{g},q,\psi ,\chi )$ and on the element $\mathbf{j}$. The following theorem establishes sufficient conditions on the problem data to ensure the regularity of the Lagrange multiplier.

Theorem 6.

Let the conditions of Theorem 5 and the inequalities (62), (63) be satisfied for all $u\in K$ and $\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$. Then, every nontrivial Lagrange multiplier $({\lambda }_0,{\mathbf{y}}^{*})$ satisfying (93)–(95) is regular, i.e., it has the form $(1,{\mathbf{y}}^{*})$ and, moreover, it is defined in a unique way.

Proof of Theorem 6.

To prove the regularity of the Lagrange multiplier, it is enough to prove that for any pair $(\widehat{\mathbf{x}},\widehat{u})\equiv (\widehat{\mathbf{u}},\widehat{\mathbf{H}},\widehat{p},\widehat{T}$, $\widehat{u})$, connected by the relation $F(\widehat{\mathbf{x}},\widehat{u},\mathbf{j})=0$, the system (93)–(95) at ${\lambda }_0=0$ has only a trivial solution. Suppose the opposite, i.e., that there exists at least one nontrivial solution ${\mathbf{y}}^{*}\equiv ((\xi ,\eta ),\sigma ,{\zeta }_1,{\zeta }_2,\theta ,{\zeta }_3)\in Y^{*}$ of the system (93)–(95) at ${\lambda }_0=0$. Then, assuming $\mathbf{w}=\xi$, $\mathbf{h}=\eta$, $r=\sigma$, $\tau =\theta$, we arrive at the equality

$$\nu a_0(\xi ,\xi )+{\nu }_1{a}_1(\eta ,\eta )+((\xi ·\nabla )\widehat{\mathbf{u}},\xi )+\kappa (\mathrm{rot}\eta \times \eta ,\widehat{\mathbf{u}})-$$

$$-\kappa (\mathrm{rot}\widehat{\mathbf{H}}\times \eta ,\xi )+\tilde{\kappa }(\xi ·\nabla \widehat{T},\theta )=0,$$

(97)

$$\tilde{\kappa }[\lambda \tilde{a}(\theta ,\theta )+\lambda {(\widehat{\alpha }\theta ,\theta )}_{{\mathsf{\Gamma }}_N}]=-b_1(\theta ,\xi ),b(\xi ,\sigma )=0.$$

(98)

Using (24), (25), (27), (28), (41) and the estimates (61) for the quadriple $(\widehat{\mathbf{u}},\widehat{\mathbf{H}},\widehat{T},\widehat{p})$, we deduce that

$$|((\xi ·\nabla )\widehat{\mathbf{u}},\xi )|\le {\gamma }_0\parallel \widehat{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}^2\le {\gamma }_0{M}_{\mathbf{u}}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}^2,$$

$$\kappa |(\mathrm{rot}\eta \times \eta ),\widehat{\mathbf{u}}|\le {\gamma }_1{M}_{\mathbf{u}}\kappa {\parallel \eta \parallel }_{1,\mathsf{\Omega }}^2,$$

$$|\kappa (\mathrm{rot}\widehat{\mathbf{H}}\times \eta ,\xi )|\le {\gamma }_1\kappa \parallel \widehat{\mathbf{H}}{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}{\parallel \eta \parallel }_{1,\mathsf{\Omega }}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}\le {\gamma }_1\kappa M_{\mathbf{H}}{\parallel \eta \parallel }_{1,\mathsf{\Omega }}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}\le$$

$$\le \frac{{\gamma }_1\sqrt{\kappa }}{2}{M}_{\mathbf{H}}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}^2+\frac{{\gamma }_1\sqrt{\kappa }}{2}{M}_{\mathbf{H}}\kappa {\parallel \eta \parallel }_{1,\mathsf{\Omega }}^2$$

$$|(\xi ·\nabla \widehat{T},\theta )|\le {\gamma }_2{\parallel \xi \parallel }_{1,\mathsf{\Omega }}\parallel \widehat{T}{\parallel }_{1,\mathsf{\Omega }}{\parallel \theta \parallel }_{1,\mathsf{\Omega }}\le {\gamma }_2{\parallel \xi \parallel }_{1,\mathsf{\Omega }}{M}_T{\parallel \theta \parallel }_{1,\mathsf{\Omega }},$$

$$|b_1(\theta ,\xi )|\le {\beta }_1{\parallel \theta \parallel }_{1,\mathsf{\Omega }}{\parallel \xi \parallel }_{1,\mathsf{\Omega }}.$$

(99)

From Lemma 2 applied to the first relation in (98), it follows that ${\parallel \parallel \theta \parallel }_{1,\mathsf{\Omega }}\le$$({\beta }_1/{\alpha }_2\lambda \tilde{\kappa }){\parallel \xi \parallel }_{1,\mathsf{\Omega }}$. Taking into account this estimate and the inequalities (99), from (97) we arrive at the following inequality

$$\left({\alpha }_0\nu -{\gamma }_0{M}_{\mathbf{u}}-\frac{{\gamma }_1\sqrt{\kappa }}{2}{M}_{\mathbf{H}}-\frac{{\beta }_1{\gamma }_2}{{\alpha }_2\lambda }{M}_T\right){\parallel \xi \parallel }_{1,\mathsf{\Omega }}^2+$$

$$+\left({\alpha }_1{\nu }_m-{\gamma }_1{M}_{\mathbf{u}}-\frac{{\gamma }_1\sqrt{\kappa }}{2}{M}_{\mathbf{H}}\right)\kappa {\parallel \eta \parallel }_{1,\mathsf{\Omega }}^2\le 0.$$

(100)

We deduce from (98), (100), considering (62), (63), that $\xi =0$, $\eta =0$, and $\theta =0$. Substituting $\xi =0,\eta =0$ and $\theta =0$ into (93), (94), we obtain

$$-(\mathrm{div}\mathbf{w},\sigma )+{\langle {\zeta }_1,\mathbf{w}\rangle }_{\mathsf{\Gamma }}+{\langle {\zeta }_2,\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}=0\forall (\mathbf{w},\mathbf{h})\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right),$$

$${\langle {\zeta }_3,\tau \rangle }_{{\mathsf{\Gamma }}_D}=0.$$

(101)

From the second identity in (101) it follows that ${\zeta }_3=0$ in $H^{1/2}{\left({\mathsf{\Gamma }}_D\right)}^{*}$. Setting $\mathbf{w}=0$ in (101), we have ${\langle {\xi }_2,\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}=0$ for all $\mathbf{h}\in {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)$. It follows from this fact that ${\zeta }_2=0$ in ${\tilde{H}}^{-s}\left(\mathsf{\Gamma }\right)$. Assuming ${\zeta }_2=0$ in the first identity in (101), we easily deduce that $\sigma =0$, ${\zeta }_1=0$ in ${\mathbf{H}}_T^{-1/2}\left(\mathsf{\Gamma }\right)$.

As a result, we shown that ${\mathbf{y}}^{*}\equiv ((\xi ,\eta ),\sigma ,{\zeta }_1,{\zeta }_2,\theta ,{\zeta }_3)=0$. This implies the regularity of the Lagrange multiplier, i.e., that ${\lambda }_0=1$. In turn, the uniqueness of the Lagrange multiplier $(1,{\mathbf{y}}^{*})$ follows from the Fredholm property of the linear operator $F_{\mathbf{x}}^{\prime }(\widehat{\mathbf{x}},\widehat{u}):$$X\to Y$. □

Below we need the partial derivatives of the Lagrangian $\mathcal{L}$ over all the controls. A simple analysis shows that

$$\langle {\mathcal{L}}_{\mathbf{g}}^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j},1,{\mathbf{y}}^{*}),\mathbf{g}\rangle ={\mu }_1{(\widehat{\mathbf{g}},\mathbf{g})}_{1/2,\mathsf{\Gamma }}-{\langle {\zeta }_1,\mathbf{g}\rangle }_{\mathsf{\Gamma }}\forall \mathbf{g}\in K_1,$$

$$\langle {\mathcal{L}}_q^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j},1,{\mathbf{y}}^{*}),q\rangle ={\mu }_2{(\widehat{q},q)}_{s,\mathsf{\Gamma }}-{\langle {\zeta }_2,q\rangle }_{\mathsf{\Gamma }}\forall q\in K_2,$$

$$\langle {\mathcal{L}}_{\psi }^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j},1,{\mathbf{y}}^{*}),\psi \rangle ={\mu }_3{(\widehat{\psi },\psi )}_{1/2,{\mathsf{\Gamma }}_D}-\tilde{\kappa }{\langle {\zeta }_3,\psi \rangle }_{{\mathsf{\Gamma }}_D}\forall \psi \in K_3,$$

$$\langle {\mathcal{L}}_{\chi }^{\prime }(\widehat{\mathbf{x}},\widehat{u},\mathbf{j},1,{\mathbf{y}}^{*}),\chi \rangle ={\mu }_4{(\widehat{\chi },\chi )}_{{\mathsf{\Gamma }}_N}-\tilde{\kappa }{\langle \chi ,\theta \rangle }_{{\mathsf{\Gamma }}_N}\forall \chi \in K_4.$$

(102)

The inequality (96) means that the quadruple $\widehat{u}=(\widehat{\mathbf{g}},\widehat{q},\widehat{\psi },\widehat{\chi })$ is the minimizer of the functional $\mathcal{L}(\widehat{\mathbf{x}},\mathbf{j},1,{\mathbf{y}}^{*})$ on the convex closed set $K=K_1\times K_2\times K_3\times K_4$. From (102) then it appears that the following variational inequalities are fulfilled:

$${\mu }_1{(\widehat{\mathbf{g}},\mathbf{g}-\widehat{\mathbf{g}})}_{1/2,\mathsf{\Gamma }}-{\langle {\zeta }_1,\mathbf{g}-\widehat{\mathbf{g}}\rangle }_{\mathsf{\Gamma }}\ge 0\forall \mathbf{g}\in K_1,$$

(103)

$${\mu }_2{(\widehat{q},q-\widehat{q})}_{s,\mathsf{\Gamma }}-{\langle {\zeta }_2,q-\widehat{q}\rangle }_{\mathsf{\Gamma }}\ge 0\forall q\in K_2,$$

(104)

$${\mu }_3{(\widehat{\psi },\psi -\widehat{\psi })}_{1/2,{\mathsf{\Gamma }}_D}-\tilde{\kappa }{\langle {\zeta }_3,\psi \rangle }_{{\mathsf{\Gamma }}_D}\ge 0\forall \psi \in K_3,$$

(105)

$${\mu }_4{(\widehat{\chi },\chi -\widehat{\chi })}_{{\mathsf{\Gamma }}_N}-\tilde{\kappa }{\langle \chi -\widehat{\chi },\theta \rangle }_{{\mathsf{\Gamma }}_N}\ge 0\forall \chi \in K_4.$$

(106)

The relations (93)–(95) together with the minimum principle (96), which is equivalent to the variational inequalities (103)–(106) with respect to controls $\widehat{\mathbf{g}}$, $\widehat{q}$, $\widehat{\psi }$ and $\widehat{\chi }$, and the operator constraint (76) constitute an optimality system. From the conditions $\xi \in {\mathbf{H}}_0^1\left(\mathsf{\Omega }\right)$, $\eta \in {\mathbf{V}}_T$, $\theta \in \mathcal{T}$ and (95) it follows that the multipliers $(\xi ,\eta ,\theta )$ adjoint of $(\mathbf{u},\mathbf{H},T)$ have the following properties:

$${\xi |}_{\mathsf{\Gamma }}=\mathbf{0},\eta ·\mathbf{n}{{|}_{\mathsf{\Gamma }}=0,\theta |}_{{\mathsf{\Gamma }}_D}=0\mathrm{and}\mathrm{div}\eta =0,\mathrm{div}\xi =-{\lambda }_0{J}_p^{\prime }(\widehat{\mathbf{x}},\widehat{u})\mathrm{i}\mathrm{n}\mathsf{\Omega }.$$

(107)

5. Additional Properties of Optimal Solutions

The next step in our study is to derive additional properties of optimal solutions, which we will need below when deriving the stability estimates. One such property is an important inequality for the difference between the solution $({\mathbf{x}}_1,u_1)$ of the problem (78) and the solution $({\mathbf{x}}_2,u_2)$ of the perturbed problem

$$\tilde{J}(\mathbf{x},u)=\frac{{\mu }_0}{2}\tilde{I}\left(\mathbf{x}\right)+\frac{{\mu }_1}{2}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\frac{{\mu }_2}{2}{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\frac{{\mu }_3}{2}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\frac{{\mu }_4}{2}{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\to inf,$$

$$F(\mathbf{x},u,\tilde{\mathbf{j}})=0,(\mathbf{x},u)\in X\times K.$$

(108)

It is obtained from (78) by replacing the functional I in (75) with a close functional $\tilde{I}$ and the function $\mathbf{j}$ by a close function $\tilde{\mathbf{j}}$. By Theorem 1, for quadruples $({\mathbf{u}}_i,{\mathbf{H}}_i,T_i,p_i)$, $i=1,2$, the following estimates hold:

$$\parallel {\mathbf{u}}_i{\parallel }_{1,\mathsf{\Omega }}\le M_{\mathbf{u}}^0=\underset{u\in K,\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)}{sup}{M}_{\mathbf{u}}(u,\mathbf{j}),$$

$$\parallel {\mathbf{H}}_i{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le M_{\mathbf{H}}^0=\underset{u\in K,\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)}{sup}{M}_{\mathbf{H}}(u,\mathbf{j}),$$

$$\parallel T_i{\parallel }_{\mathsf{\Omega }}\le M_T^0=\underset{u\in K,\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)}{sup}{M}_T(u,\mathbf{j}),{\parallel p_i\parallel }_{\mathsf{\Omega }}\le M_p^0=\underset{u\in K,\mathbf{j}\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)}{sup}{M}_p(u,\mathbf{j}).$$

(109)

We assume that the following conditions hold:

$$({\gamma }_0/{\alpha }_0\nu )M_{\mathbf{u}}^0+({\gamma }_1/{\alpha }_0\nu )(\sqrt{\kappa }/2)M_{\mathbf{H}}^0+({\beta }_1{\gamma }_2/{\alpha }_0\nu {\alpha }_2\lambda )M_T^0<1/2,$$

$$({\gamma }_1/{\alpha }_1{\nu }_m)M_{\mathbf{u}}^0+({\gamma }_1/{\alpha }_1{\nu }_m)(\sqrt{\kappa }/2)M_{\mathbf{H}}^0<1/2.$$

(110)

To make conditions (110) more illustrative and to simplify the further presentation let us define the parameters

$$\mathcal{R}e={\gamma }_0{M}_{\mathbf{u}}^0/\left({\alpha }_0\nu \right),\mathcal{R}m={\gamma }_1{M}_{\mathbf{u}}^0/\left({\alpha }_1{\nu }_m\right),\mathcal{R}a=\frac{{\beta }_1{\gamma }_2{M}_T^0}{{\alpha }_0\nu {\alpha }_2\lambda },$$

$$\mathcal{H}a={\gamma }_1\sqrt{\kappa }{M}_{\mathbf{H}}^0/\left({\alpha }_0\nu \right),\mathcal{P}m={\alpha }_0\nu /\left({\alpha }_1{\nu }_m\right).$$

(111)

They are close to the following dimensionless parameters, which are widely used in hydrodynamics (see the textbook by Shercliff [11]): the Reynolds number Re, the magnetic Reynolds number Rm, the Raley number Ra, the Hartman number Ha and the magnetic Prandtl number Pm. We emphasize that parameters $\mathcal{R}e$, $\mathcal{R}m$, $\mathcal{R}a$, $\mathcal{H}a$ and $\mathcal{P}m$ are dimensionless. To prove this fact, it is enough to know the dimensions of parameters ${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$, ${\gamma }_0,{\gamma }_1$, ${\gamma }_2$, ${\beta }_1$ and also the dimensions of $M_{\mathbf{u}}^0$, $M_T^0$ and $M_{\mathbf{H}}^0$ included in (110). Their dimensions were given in (39).

Using (39), we can easily prove that all the parameters $\mathcal{R}e$, $\mathcal{R}m$, $\mathcal{R}a$, $\mathcal{H}a$ and $\mathcal{P}m$ defined in (111) are dimensionless. We emphasize that parameters $\mathcal{R}e$ and $\mathcal{R}m$ are related by the relation $\mathcal{R}m=({\gamma }_1/{\gamma }_0)\mathcal{P}m\mathcal{R}e$. Therefore, we can rewrite conditions (110) in a simpler form containing only four dimensionless parameters $\mathcal{R}e$, $\mathcal{H}a$, $\mathcal{P}m$ and $\mathcal{R}a$:

$$\mathcal{R}e+(1/2)\mathcal{H}a+\mathcal{R}a<1/2,({\gamma }_1/{\gamma }_0)\mathcal{P}m\mathcal{R}e+(1/2)\mathcal{P}m\mathcal{H}a<1/2.$$

(112)

We denote by $(1,{\mathbf{y}}_i^{*})\equiv (1,({\xi }_i,{\eta }_i),{\sigma }_i,{\zeta }_1^{\left(i\right)},{\zeta }_2^{\left(i\right)},{\theta }_i,{\zeta }_3^{\left(i\right)})$, $i=1,2$, the Lagrange multipliers corresponding to solutions $({\mathbf{x}}_1,u_1)$ and $({\mathbf{x}}_2,u_2)$ of the problems (78) and (108), respectively, (these multipliers are uniquely determined under conditions (112)). By definition, they satisfy identities

$$\nu (\nabla \mathbf{w},\nabla {\xi }_i)+{\nu }_1(\mathrm{rot}\mathbf{h},\mathrm{rot}{\eta }_i)+(({\mathbf{u}}_i·\nabla )\mathbf{w},{\xi }_i)+((\mathbf{w}·\nabla ){\mathbf{u}}_i,{\xi }_i)+$$

$$+\kappa [(\mathrm{rot}{\eta }_i\times {\mathbf{H}}_i,\mathbf{w})+(\mathrm{rot}{\eta }_i\times \mathbf{h},{\mathbf{u}}_i)]-$$

$$-\kappa [(\mathrm{rot}{\mathbf{H}}_i\times \mathbf{h},{\xi }_i)+(\mathrm{rot}\mathbf{h}\times {\mathbf{H}}_i,{\xi }_i)]+\kappa (\mathbf{w}·\nabla T_i,{\theta }_i)-$$

$$-(\mathrm{div}\mathbf{w},{\sigma }_i)+{\langle {\zeta }_1^{\left(i\right)},\mathbf{w}\rangle }_{\mathsf{\Gamma }}+$$

$$+{\langle {\zeta }_2^{\left(i\right)},\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}=-({\mu }_0/2)\langle {\left(I^i\right)}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_i\right),\mathbf{w}\rangle -$$

$$-({\mu }_0/2)\langle {\left(I^i\right)}_{\mathbf{H}}^{\prime }\left({\mathbf{x}}_i\right),\mathbf{h}\rangle \forall (\mathbf{w},\mathbf{h})\in {\mathbf{H}}_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right),i=1,2,$$

(113)

$$\tilde{\kappa }[\lambda \tilde{a}(\tau ,{\theta }_i)+\lambda {(\alpha \tau ,{\theta }_i)}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_i·\nabla \tau ,{\theta }_i)+{\langle {\zeta }_3^{\left(i\right)},\tau \rangle }_{{\mathsf{\Gamma }}_D}]+b_1(\tau ,{\xi }_i)=$$

$$=-({\mu }_0/2)\langle {\left(I^i\right)}_T^{\prime }({\mathbf{x}}_i,u_i),\tau \rangle \forall \tau \in H^1\left(\mathsf{\Omega }\right),$$

(114)

$$(\mathrm{div}{\xi }_i,r)=({\mu }_0/2)({\left(I^i\right)}_p^{\prime }\left({\mathbf{x}}_i\right),r)\forall r\in L_0^2\left(\mathsf{\Omega }\right),i=1,2.$$

(115)

Here, we changed the notations as $I=I^1$, $\tilde{I}=I^2$. Let us define the following differences:

$$\mathbf{u}={\mathbf{u}}_1-{\mathbf{u}}_2,\mathbf{H}={\mathbf{H}}_1-{\mathbf{H}}_2,T=T_1-T_2,p=p_1-p_2,$$

$$\mathbf{j}={\mathbf{j}}_1-{\mathbf{j}}_2,\mathbf{g}={\mathbf{g}}_1-{\mathbf{g}}_2,q=q_1-q_2,$$

$$\xi ={\xi }_1-{\xi }_2,\eta ={\eta }_1-{\eta }_2,\theta ={\theta }_1-{\theta }_2,\sigma ={\sigma }_1-{\sigma }_2,$$

$${\zeta }_1={\zeta }_1^{\left(1\right)}-{\zeta }_1^{\left(2\right)},{\zeta }_2={\zeta }_2^{\left(1\right)}-{\zeta }_2^{\left(2\right)},{\zeta }_3={\zeta }_3^{\left(1\right)}-{\zeta }_3^{\left(2\right)}.$$

(116)

Now, reasoning as in Alekseev [58], we derive an important inequality for differences (116), which will be used in Section 6 to prove the stability estimates for optimal solutions. While deriving the mentioned inequality we will use some ideas and results from Alekseev et al. [23], which we sketch here for the reader’s convenience.

Firstly, we subtract relations (54)–(56), written for ${\mathbf{u}}_2$, ${\mathbf{H}}_2$, $T_2$, $p_2$, $u_2$, ${\mathbf{j}}_2$, from (54)–(56) written for ${\mathbf{u}}_1$, ${\mathbf{H}}_1$, $T_1$, $p_1$, $u_1$, ${\mathbf{j}}_1$. Using (116), we obtain

$$\nu (\nabla \mathbf{u},\nabla \mathbf{v})+{\nu }_1(\mathrm{rot}\mathbf{H},\mathrm{rot}\mathbf{\Psi })+[((\mathbf{u}·\nabla ){\mathbf{u}}_1,\mathbf{v})+(({\mathbf{u}}_2·\nabla )\mathbf{u},\mathbf{v})]+$$

$$+\kappa [(\mathrm{rot}\mathbf{\Psi }\times \mathbf{H},{\mathbf{u}}_1)+(\mathrm{rot}\mathbf{\Psi }\times {\mathbf{H}}_2,\mathbf{u})]-$$

$$-\kappa [(\mathrm{rot}{\mathbf{H}}_1\times \mathbf{H},\mathbf{v})+(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}_2,\mathbf{v})]-(\mathrm{div}\mathbf{v},p)+b_1(T,\mathbf{v})=$$

$$={\nu }_1(\mathbf{j},\mathrm{rot}\mathbf{\Psi })\forall \left(\mathbf{v},\mathbf{\Psi }\right)\in H_0^1{\left(\mathsf{\Omega }\right)}^3\times V_T,$$

(117)

$$\lambda \tilde{a}(T,S)+\lambda {(\alpha T,S)}_{{\mathsf{\Gamma }}_N}+((\mathbf{u}·\nabla )T_1,S)+({\mathbf{u}}_2·\nabla T,S)={(\chi ,S)}_{{\mathsf{\Gamma }}_N}\forall S\in \mathcal{T},$$

(118)

$${\mathrm{div}\mathbf{u}=0\mathrm{i}\mathrm{n}\mathsf{\Omega },\mathbf{u}|}_{\mathsf{\Gamma }}=\mathbf{g},\mathbf{H}·\mathbf{n}{{|}_{\mathsf{\Gamma }}=q,T|}_{{\mathsf{\Gamma }}_D}=\psi .$$

(119)

Then, let us substitute $\mathbf{g}={\mathbf{g}}_1$ into the inequality (103) written at $\widehat{\mathbf{g}}={\mathbf{g}}_2$, ${\zeta }_1={\zeta }_1^{\left(2\right)}$, and then substitute $\mathbf{g}={\mathbf{g}}_2$ into (103) written at $\widehat{\mathbf{g}}={\mathbf{g}}_1$, ${\zeta }_1={\zeta }_1^{\left(1\right)}$. We obtain

$${\mu }_1{({\mathbf{g}}_2,\mathbf{g})}_{1/2,\mathsf{\Gamma }}+{\langle {\zeta }_1^{\left(2\right)},\mathbf{g}\rangle }_{\mathsf{\Gamma }}\ge 0,-{\mu }_1{({\mathbf{g}}_1,\mathbf{g})}_{1/2,\mathsf{\Gamma }}+{\langle {\zeta }_1^{\left(1\right)},\mathbf{g}\rangle }_{\mathsf{\Gamma }}\ge 0.$$

Adding these inequalities, we obtain the relation

$$-{\langle {\zeta }_1,\mathbf{g}\rangle }_{\mathsf{\Gamma }}\le -{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2.$$

(120)

Using the same scheme, from (104)–(106) we derive the inequalities

$$-{\langle {\zeta }_2,q\rangle }_{\mathsf{\Gamma }}\le -{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2,$$

$$-\tilde{\kappa }{\langle {\zeta }_3,\psi \rangle }_{{\mathsf{\Gamma }}_D}\le -{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2,-\tilde{\kappa }{(\chi ,\theta )}_{{\mathsf{\Gamma }}_N}\le -{\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2.$$

(121)

Further, we subtract identities (113)–(115) for $i=2$ from (113)–(115) for $i=1$ and set $\mathbf{w}=\mathbf{u},\mathbf{h}=\mathbf{H},\tau =T,r=p$. Adding up the obtained results and using relations ${\mathbf{u}|}_{\mathsf{\Gamma }}=\mathbf{g}$, ${\mathbf{H}·\mathbf{n}|}_{\mathsf{\Gamma }}=q$, ${T|}_{{\mathsf{\Gamma }}_D}=\psi$ we obtain

$$\nu (\nabla \mathbf{u},\nabla \xi )+{\nu }_1(\mathrm{rot}\mathbf{H},\mathrm{rot}\eta )+(({\mathbf{u}}_1·\nabla )\mathbf{u},\xi )+2((\mathbf{u}·\nabla )\mathbf{u},{\xi }_2)+((\mathbf{u}·\nabla ){\mathbf{u}}_1,\xi )+$$

$$+\kappa [(\mathrm{rot}\eta \times {\mathbf{H}}_1,\mathbf{u})+2(\mathrm{rot}{\eta }_2\times \mathbf{H},\mathbf{u})+(\mathrm{rot}\eta \times \mathbf{H},{\mathbf{u}}_1)]-$$

$$-\kappa [(\mathrm{rot}{\mathbf{H}}_1\times \mathbf{H},\xi )+2(\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_2)+(\mathrm{rot}{\mathbf{H}\times \mathbf{H}}_1,\xi )]-(\mathrm{div}\xi ,p)+$$

$$+\tilde{\kappa }(\mathbf{u}·\nabla T_1,\theta )+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_2)+$$

$$\tilde{\kappa }\left[\lambda \tilde{a}(T,\theta )+\lambda {(\alpha T,\theta )}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_1·\nabla T,\theta )+(\mathbf{u}·\nabla T,{\theta }_2)+{\langle {\zeta }_3,\psi \rangle }_{{\mathsf{\Gamma }}_D}\right]+$$

$$+(\mathbf{b}T,\xi )=-{\langle {\zeta }_1,\mathbf{g}\rangle }_{\mathsf{\Gamma }}-{\langle {\zeta }_2,q\rangle }_{\mathsf{\Gamma }}-\tilde{\kappa }{\langle {\zeta }_3,\psi \rangle }_{{\mathsf{\Gamma }}_D}-$$

$$-({\mu }_0/2)\langle I_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_2\right),\mathbf{u}\rangle -({\mu }_0/2)\langle I_{\mathbf{H}}^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_{\mathbf{H}}^{\prime }\left({\mathbf{x}}_2\right),\mathbf{H}\rangle -$$

$$-({\mu }_0/2)\langle I_p^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_p^{\prime }\left({\mathbf{x}}_2\right),p\rangle -({\mu }_0/2)\tilde{\kappa }\langle I_T^{\prime }\left({\mathbf{x}}_1\right)-I_T^{\prime }\left({\mathbf{x}}_2\right),T\rangle .$$

(122)

Now, we set $\mathbf{v}=\xi$, $\mathbf{\Psi }=\eta$ in (117) and $S=\tilde{\kappa }\theta$ in (118). We obtain

$$\nu (\nabla \mathbf{u},\nabla \xi )+{\nu }_1(\mathrm{rot}\mathbf{H},\mathrm{rot}\eta )+[((\mathbf{u}·\nabla ){\mathbf{u}}_1,\xi )+(({\mathbf{u}}_2·\nabla )\mathbf{u},\xi )]+$$

$$+\kappa [(\mathrm{rot}\eta \times \mathbf{H},{\mathbf{u}}_1)+(\mathrm{rot}\eta \times {\mathbf{H}}_2,\mathbf{u})]-$$

$$-\kappa [(\mathrm{rot}{\mathbf{H}}_1\times \mathbf{H},\xi )+(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}_2,\xi )]-(\mathrm{div}\mathbf{v},p)+b_1(T,\xi )={\nu }_1(\mathbf{j},\mathrm{rot}\eta ),$$

(123)

$$\tilde{\kappa }[\lambda \tilde{a}(T,\theta )+\lambda {(\alpha T,\theta )}_{{\mathsf{\Gamma }}_N}+((\mathbf{u}·\nabla )T_1,\theta )+({\mathbf{u}}_2·\nabla T,\theta )]=\tilde{\kappa }{(\chi ,\theta )}_{{\mathsf{\Gamma }}_N}.$$

(124)

Finally, we add (123) to (124) and subtract the obtained result from (122). Using (120), (121) and identities

$$2((\mathbf{u}·\nabla )\mathbf{u},{\xi }_2)+(({\mathbf{u}}_1·\nabla )\mathbf{u},\xi )-(({\mathbf{u}}_2·\nabla )\mathbf{u},\xi )=$$

$$=2((\mathbf{u}·\nabla )\mathbf{u},{\xi }_2)+((\mathbf{u}·\nabla )\mathbf{u},\xi )=((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2),$$

$$-2(\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_2)+(\mathrm{rot}{\mathbf{H}\times \mathbf{H}}_2,\xi )-(\mathrm{rot}{\mathbf{H}\times \mathbf{H}}_1,\xi )=-(\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2),$$

$$2(\mathrm{rot}{\eta }_2\times \mathbf{H},\mathbf{u})+(\mathrm{rot}\eta \times {\mathbf{H}}_1,\mathbf{u})-(\mathrm{rot}\eta \times {\mathbf{H}}_2,\mathbf{u})=(\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u}),$$

$$2(\mathbf{u}·\nabla T,{\theta }_2)+(\mathbf{u}·\nabla T_1,\theta )-(\mathbf{u}·\nabla T_2,\theta )=(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2),$$

we obtain the inequality

$$((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)+\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})+$$

$$+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)+({\mu }_0/2)\langle I_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_2\right),\mathbf{u}\rangle +({\mu }_0/2)\langle I_{\mathbf{H}}^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_{\mathbf{H}}^{\prime }\left({\mathbf{x}}_2\right),\mathbf{H}\rangle +$$

$$+\tilde{\kappa }({\mu }_0/2)\langle I_T^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_T^{\prime }\left({\mathbf{x}}_2\right),T\rangle +({\mu }_0/2)(I_p^{\prime }\left({\mathbf{x}}_1\right)-{\tilde{I}}_p^{\prime }\left({\mathbf{x}}_2\right),p)\le$$

$$\le -{\nu }_1(\mathbf{j},\mathrm{rot}\eta )-{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-{\mu }_3{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2.$$

(125)

It is this inequality that will play an important role in Section 6 in the derivation of local uniqueness and stability of optimal solutions. Let us formulate the obtained result as the following theorem.

Theorem 7.

Let under conditions of Theorem 3 for cost functionals I and $\tilde{I}$ and under conditions (112), pairs $({\mathbf{x}}_1,u_1)=({\mathbf{u}}_1$, ${\mathbf{H}}_1$, $T_1$, $p_1$, ${\mathbf{g}}_1$, $q_1$, ${\psi }_1$, ${\chi }_1)\in X\times K$ and $({\mathbf{x}}_2,u_2)=({\mathbf{u}}_2$, ${\mathbf{H}}_2$, $T_2$, $p_2$, ${\mathbf{g}}_2$, $q_2$, ${\psi }_2$, ${\chi }_2)\in X\times K$ be solutions to the problems (78) and (108), respectively. Let ${\mathbf{y}}_i^{*}=(({\xi }_i,{\eta }_i)$, ${\sigma }_i$, ${\zeta }_1^{\left(i\right)}$, ${\zeta }_2^{\left(i\right)}$, ${\theta }_i$, ${\zeta }_3^{\left(i\right)})\in Y^{*}$, $i=1,2$, be the adjoint states corresponding to these solutions. Then for the differences $\mathbf{u}$, $\mathbf{H}$, T, p, $\mathbf{g}$, q, ψ, χ and $\mathbf{j}$ defined in (116), the relation (125) holds.

Now, we consider the problem (117)–(119) with respect to differences $\mathbf{u}={\mathbf{u}}_1-{\mathbf{u}}_2$, $\mathbf{H}={\mathbf{H}}_1-{\mathbf{H}}_2$, $T=T_1-T_2$ and $p=p_1-p_2$ in which differences $\mathbf{g}={\mathbf{g}}_1-{\mathbf{g}}_2$, $q=q_1-q_2$, $\psi ={\psi }_1-{\psi }_2$, $\chi ={\chi }_1-{\chi }_2$ and $\mathbf{j}={\mathbf{j}}_1-{\mathbf{j}}_2$ together with functions ${\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{H}}_1$, ${\mathbf{H}}_2$, $T_1$ and $T_2$ play the role of data.

Below we will need estimates for norms of the differences $\mathbf{u}$, $\mathbf{H}$, T and p via norms of the differences $\mathbf{g}$, q, $\psi$, $\chi$ and $\mathbf{j}$. In order to derive these estimates we represent the differences $\mathbf{u}={\mathbf{u}}_1-{\mathbf{u}}_2$, $\mathbf{H}={\mathbf{H}}_1-{\mathbf{H}}_2$ and $T=T_1-T_2$ in the form

$$\mathbf{u}={\mathbf{u}}_0+\tilde{\mathbf{u}},\mathbf{H}={\mathbf{H}}_0+\tilde{\mathbf{H}},T=T_0+\tilde{T}.$$

(126)

Here, ${\mathbf{u}}_0$ and ${\mathbf{H}}_0$ are liftings of the differences $\mathbf{g}$ and q while $\tilde{\mathbf{u}}\in \mathbf{V}$, $\tilde{\mathbf{H}}\in {\mathbf{V}}_T$ and $\tilde{T}\in \mathcal{T}$ are certain functions, respectively. It should be noted that the mentioned liftings satisfy conditions (58), (59). As to $T_0$ we will use the standard continuation $T_0={\left(\gamma {|}_{{\mathsf{\Gamma }}_D}\right)}_r^{-1}\psi$ for which the estimate (9) holds. We remind that the following estimates are valid for ${\mathbf{u}}_0$, ${\mathbf{H}}_0$ and $T_0$:

$$\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}\le C_{\epsilon }{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }},\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\equiv \parallel {\mathbf{H}}_0{\parallel }_{r,\mathsf{\Omega }}\le C_{\mathsf{\Gamma }}^s{\parallel q\parallel }_{s,\mathsf{\Gamma }},$$

$$\parallel T_0{\parallel }_{1,\mathsf{\Omega }}\le C_D{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D},$$

(127)

We set in (117), (118) the following:

$$\mathbf{u}={\mathbf{u}}_0+\tilde{\mathbf{u}},\mathbf{v}+\tilde{\mathbf{u}},\mathbf{H}={\mathbf{H}}_0+\tilde{\mathbf{H}},\mathbf{\Psi }=\tilde{\mathbf{H}},T=T_0+\tilde{T}\mathrm{and}S=\tilde{T}.$$

(128)

Taking into account (128) the relation (117) takes the form

$$\nu (\nabla \tilde{\mathbf{u}},\nabla \tilde{\mathbf{u}})+{\nu }_1(\mathrm{rot}\tilde{\mathbf{H}},\mathrm{rot}\tilde{\mathbf{H}})=-\nu (\nabla {\mathbf{u}}_0,\nabla \tilde{\mathbf{u}})-$$

$$-[((\tilde{\mathbf{u}}·\nabla ){\mathbf{u}}_1,\tilde{\mathbf{u}})+(({\mathbf{u}}_0·\nabla ){\mathbf{u}}_1,\tilde{\mathbf{u}})+(({\mathbf{u}}_2·\nabla ){\mathbf{u}}_0,\tilde{\mathbf{u}})]-$$

$$-\kappa [(\mathrm{rot}\tilde{\mathbf{H}}\times {\mathbf{H}}_0,{\mathbf{u}}_1)+(\mathrm{rot}\tilde{\mathbf{H}}\times \tilde{\mathbf{H}},{\mathbf{u}}_1)+(\mathrm{rot}\tilde{\mathbf{H}}\times {\mathbf{H}}_2,{\mathbf{u}}_0)]+$$

$$+\kappa [(\mathrm{rot}{\mathbf{H}}_1\times {\mathbf{H}}_0,\tilde{\mathbf{u}})+(\mathrm{rot}{\mathbf{H}}_1\times \tilde{\mathbf{H}},\tilde{\mathbf{u}})]-(\mathbf{b}(T_0+\tilde{T}),\tilde{\mathbf{u}})-{\nu }_1(\mathbf{j},\mathrm{rot}\tilde{\mathbf{H}})$$

(129)

while (118) transforms to

$$\lambda \tilde{a}(\tilde{T},\tilde{T})+\lambda {(\alpha \tilde{T},\tilde{T})}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_2·\nabla \tilde{T},\tilde{T})=-\lambda \tilde{a}(T_0,\tilde{T})-$$

$$-\lambda {(\alpha T_0,\tilde{T})}_{{\mathsf{\Gamma }}_N}-(\tilde{\mathbf{u}}·\nabla T_1,\tilde{T})-({\mathbf{u}}_0·\nabla T_1,\tilde{T})-({\mathbf{u}}_2·\nabla T_0,\tilde{T})+{(\chi ,\tilde{T})}_{{\mathsf{\Gamma }}_N}.$$

(130)

Using (23), (27), (31), (32) and the estimates (127), we have consistently

$$|\lambda \tilde{a}(T_0,\tilde{T})|\le \lambda \parallel T_0{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le C_D{\lambda \parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }},$$

$$|\lambda (\alpha T_0,\tilde{T})|\le C_N{\lambda \parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}\parallel T_0{\parallel }_{1,\mathsf{\Omega }}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }}\le$$

$$\le C_N{C}_D{\lambda \parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }},$$

$$|(\tilde{\mathbf{u}}·\nabla T_1,\tilde{T})|\le {\gamma }_2\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\parallel T_1{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le {\gamma }_2{M}_T^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }},$$

$$|({\mathbf{u}}_0·\nabla T_1,\tilde{T})|\le {\gamma }_2\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}\parallel T_1{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le {\gamma }_2{C}_{{\epsilon }_0}{M}_T^0{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }},$$

$$|{\mathbf{u}}_2·\nabla T_0,\tilde{T}|\le {\gamma }_2\parallel {\mathbf{u}}_2{\parallel }_{1,\mathsf{\Omega }}\parallel T_0{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le {\gamma }_2{M}_{\mathbf{u}}^0{C}_D{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }},$$

$$|{(\chi ,\tilde{T})}_{{\mathsf{\Gamma }}_N}|\le {\gamma }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}{\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }}.$$

Moreover, it follows from Lemma 2 that

$$\lambda \tilde{a}(\tilde{T},\tilde{T})+\lambda {(\alpha \tilde{T},\tilde{T})}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_2·\nabla \tilde{T},\tilde{T})\ge {\alpha }_2\lambda {\parallel \tilde{T}\parallel }_{1,\mathsf{\Omega }}^2\forall \tilde{T}\in \mathcal{T}.$$

Taking this into account, from (130) we arrive at the inequality

$$\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le {\left({\alpha }_2\lambda \right)}^{-1}\left[\right(C_D\lambda +C_D{C}_N{\lambda \parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}+{\gamma }_2{C}_D{M}_{\mathbf{u}}^0{)\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+$$

$$+{\gamma }_2{C}_{{\epsilon }_0}{M}_T^0{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+{\gamma }_2{M}_T^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+{\gamma }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}].$$

It follows from it and from the third estimate in (127) that

$$\parallel T_0+\tilde{T}{\parallel }_{1,\mathsf{\Omega }}\le d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+d_2\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}.$$

(131)

Here,

$$d_1={\left({\alpha }_2\lambda \right)}^{-1}\left({\gamma }_2{C}_{{\epsilon }_0}{M}_T^0\right),d_2={\left({\alpha }_2\lambda \right)}^{-1}\left({\gamma }_2{M}_T^0\right),$$

$$d_3=C_D[1+{\left({\alpha }_2\lambda \right)}^{-1}(\lambda +C_N{\lambda \parallel \alpha \parallel }_{{\mathsf{\Gamma }}_N}+{\gamma }_2{M}_{\mathbf{u}}^0\left)\right],$$

$$d_4={\left({\alpha }_2\lambda \right)}^{-1}{\gamma }_4.$$

(132)

Considering (131), we have

$$\parallel T_0+\tilde{T}{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+d_2{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}.$$

(133)

Using estimates (21), (22), (24), (25), (28), (41), estimates (109), and setting ${\nu }_1={\nu }_m\kappa$ one can obtain from (129) that

$${\alpha }_0\nu \parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+{\alpha }_1{\nu }_m\kappa \parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2\le \nu \parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}+$$

$$+{\gamma }_0{M}_{\mathbf{u}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+2{\gamma }_0{M}_{\mathbf{u}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}+$$

$$+\kappa {\gamma }_1{M}_{\mathbf{u}}^0\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+\kappa {\gamma }_1{M}_{\mathbf{u}}^0\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}+$$

$$+\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}{\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}+$$

$$+{\beta }_1\parallel T_0{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+{\beta }_1\parallel \tilde{T}{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}+C_0{\nu }_m{\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}.$$

(134)

Applying Young’s inequality of one of the following types to the summands in (134):

$$|ab|\le \frac{\epsilon a^2}{2}+\frac{b^2}{2\epsilon }\mathrm{o}\mathrm{r}2ab\le \epsilon \frac{a^2}{2}+\frac{2b^2}{\epsilon },\epsilon =\mathrm{const},$$

(135)

for $\epsilon ={\alpha }_0\nu /6$ or $\epsilon ={\alpha }_0\nu /12$ or $\epsilon ={\alpha }_1{\nu }_m\kappa /6$ or $\epsilon =1$, we consistently derive

$$2{\gamma }_0{M}_{\mathbf{u}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{2{\left({\gamma }_0{M}_{\mathbf{u}}^0\right)}^2}{\epsilon }{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2=$$

$$=\frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+24{\alpha }_0\nu \mathcal{R}{e}^2{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2,$$

(136)

$${\gamma }_1\kappa M_{\mathbf{u}}^0\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{{\left({\gamma }_1\kappa M_{\mathbf{u}}^0\right)}^2}{2\epsilon }{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2=$$

$$=\frac{{\alpha }_1{\nu }_m\kappa }{12}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+3\kappa {\alpha }_0\nu \mathcal{P}m{\left(\frac{{\gamma }_1}{{\gamma }_0}\right)}^2\mathcal{R}{e}^2{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2,$$

(137)

$${\gamma }_1\kappa M_{\mathbf{H}}^0\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{{\left({\gamma }_1\kappa M_{\mathbf{H}}^0\right)}^2}{2\epsilon }{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2=$$

$$=\frac{{\alpha }_1{\nu }_m\kappa }{12}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+3{\alpha }_0\nu \mathcal{P}m\mathcal{H}{a}^2{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2,$$

(138)

$$\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}{\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}\le (\sqrt{\kappa }/2){\gamma }_1{M}_{\mathbf{H}}^0\left(\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\kappa {\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}^2\right),$$

(139)

$$\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le \frac{\epsilon }{2}{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}^2+\frac{(\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel {\mathbf{H}}_0{{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)})}^2}{2\epsilon }=$$

$$=\frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+6\kappa {\alpha }_0\nu \mathcal{H}{a}^2{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2,$$

(140)

$$\nu \parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}\le \frac{\nu \epsilon }{2}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{\nu }{2\epsilon }{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$\le \frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{6\nu }{{\alpha }_0}{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2,$$

(141)

$$C_0{\nu }_m{\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\le \frac{{\alpha }_1{\nu }_m\kappa }{12}\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{3C_0^2{\nu }_m\kappa }{{\alpha }_1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2.$$

(142)

By a similar scheme, using the first inequality in (135) at $\epsilon ={\alpha }_0\nu /12$, we derive

$${\beta }_1{d}_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{{\beta }_1^2{d}_1^2}{2\epsilon }{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2=$$

$$=\frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{6{\beta }_1^2{d}_1^2}{{\alpha }_0\nu }{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2,$$

$${\beta }_1{d}_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{{\beta }_1^2{d}_3^2}{2\epsilon }{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2=$$

$$=\frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{6{\beta }_1^2{d}_3^2}{{\alpha }_0\nu }{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2,$$

$${\beta }_1{d}_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le \frac{\epsilon }{2}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{{\beta }_1^2{d}_4^2}{2\epsilon }{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2=$$

$$=\frac{{\alpha }_0\nu }{24}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+\frac{6{\beta }_1^2{d}_4^2}{{\alpha }_0\nu }{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2,$$

$${\beta }_1{d}_2\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2={\beta }_1\frac{{\gamma }_2{M}_T}{{\alpha }_2\lambda }\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2={\alpha }_0\nu \mathcal{R}a{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}^2.$$

Using the above inequalities, we conclude that

$${\beta }_1\parallel T_0+\tilde{T}{\parallel }_{1,\mathsf{\Omega }}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le \frac{{\alpha }_0\nu }{8}\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+{\alpha }_0\nu \mathcal{R}a{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+\frac{6{\beta }_1^2}{{\alpha }_0\nu }\left(d_1^2{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+d_3^2{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+d_4^2{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\right).$$

(143)

Taking into account (136)–(142), from (134) we obtain that

$$({\alpha }_0\nu -{\gamma }_0{M}_{\mathbf{u}}^0+{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0){\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+({\alpha }_1{\nu }_m-{\gamma }_1{M}_{\mathbf{u}}^0+{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0)\kappa {\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$({\alpha }_0\nu /8)\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+({\alpha }_1{\nu }_m\kappa /4)\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2+3{\alpha }_0\nu (8\mathcal{R}{e}^2+\mathcal{P}m\mathcal{H}{a}^2+2{\alpha }_0^{-2}){\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2+$$

$$3\kappa {\alpha }_0\nu (2\mathcal{H}{a}^2+\mathcal{P}m{({\gamma }_1/{\gamma }_0)}^2\mathcal{R}{e}^2)\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2+3C_0^2{\nu }_m\kappa {\alpha }_1^{-1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+{\beta }_1{\parallel T_0+\tilde{T}\parallel }_{1,\mathsf{\Omega }}.$$

(144)

It follows from (110) that

$$({\alpha }_0\nu /2)<{\alpha }_0\nu -{\gamma }_0{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0-\frac{{\beta }_1{\gamma }_2}{{\alpha }_1\lambda }{M}_T^0,$$

$$({\alpha }_1{\nu }_m/2)<{\alpha }_1{\nu }_m-{\gamma }_1{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0.$$

(145)

Using (145), from (143) and (144) we derive that

$${\alpha }_0\nu \parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2+{\alpha }_1{\nu }_m\kappa {\parallel \tilde{\mathbf{H}}\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$12{\alpha }_0\nu (8\mathcal{R}{e}^2+\mathcal{P}m\mathcal{H}{a}^2+2{\alpha }_0^{-2}){\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2+$$

$$12\kappa {\alpha }_0\nu (2\mathcal{H}{a}^2+\mathcal{P}m{({\gamma }_1/{\gamma }_0)}^2\mathcal{R}{e}^2)\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2+12C_0^2{\nu }_m\kappa {\alpha }_1^{-1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+$$

$$+\frac{24{\beta }_1^2}{{\alpha }_0\nu }(d_1^2{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+d_3^2{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+d_4^2{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2)\le$$

$$\le {\alpha }_0\nu {\mathcal{R}}_1^2\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}^2+\kappa {\alpha }_0\nu {\mathcal{R}}_2^2\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2+12C_0^2{\nu }_m\kappa {\alpha }_1^{-1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+\frac{24{\beta }_1^2{d}^2}{{\alpha }_0\nu }.$$

(146)

Here, for brevity we introduced the notations

$${\mathcal{R}}_1^2=(96\mathcal{R}{e}^2+12\mathcal{P}m\mathcal{H}{a}^2+24{\alpha }_0^{-2}),$$

$${\mathcal{R}}_2^2=(24\mathcal{H}{a}^2+12\mathcal{P}m{({\gamma }_1/{\gamma }_0)}^2\mathcal{R}{e}^2),$$

(147)

$$d=d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}.$$

(148)

From (146) we further deduce that

$$\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}^2\le {\mathcal{R}}_1^2\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}^2+\kappa {\mathcal{R}}_2^2{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2+$$

$$+12C_0^2\frac{{\nu }_m}{\nu }\kappa \frac{{\alpha }_1}{{\alpha }_0}{\alpha }_1^{-2}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+\frac{24{\beta }_1^2{d}^2}{{\left({\alpha }_0\nu \right)}^2},$$

(149)

$$\kappa \parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}^2\le \frac{{\alpha }_0\nu }{{\alpha }_1{\nu }_m}{\mathcal{R}}_1^2{\parallel {\mathbf{u}}_0\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+\kappa \frac{{\alpha }_0\nu }{{\alpha }_1{\nu }_m}{\mathcal{R}}_2^2\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2+12C_0^2\kappa {\alpha }_1^{-2}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+\frac{24{\beta }_1^2{d}^2}{{\alpha }_0\nu {\alpha }_1{\nu }_m}.$$

(150)

It follows from (149) and (150) that

$$\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}\le {\mathcal{R}}_1\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}+\sqrt{\kappa }{\mathcal{R}}_2\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+\sqrt{12}\frac{C_0}{{\alpha }_1}\sqrt{\frac{\kappa }{\mathcal{P}m}}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\frac{\sqrt{24}{\beta }_{1}d}{{\alpha }_0\nu },$$

(151)

$$\sqrt{\kappa }\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\le \sqrt{\mathcal{P}m}{\mathcal{R}}_1\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}+\sqrt{\kappa \mathcal{P}m}{\mathcal{R}}_2{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+$$

$$+\sqrt{12}\frac{C_0}{{\alpha }_1}\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\frac{\sqrt{24}{\beta }_{1}d}{\sqrt{{\alpha }_0\nu {\alpha }_1{\nu }_m}}.$$

The last estimate can be rewritten as

$$\parallel \tilde{\mathbf{H}}{\parallel }_{1,\mathsf{\Omega }}\le \sqrt{\frac{\mathcal{P}m}{\kappa }}{\mathcal{R}}_1\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}+\sqrt{\mathcal{P}m}{\mathcal{R}}_2{\parallel {\mathbf{H}}_0\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+$$

$$+\sqrt{12}\frac{C_0}{{\alpha }_1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\frac{\sqrt{24}{\beta }_{1}d}{\sqrt{\kappa {\alpha }_0\nu {\alpha }_1{\nu }_m}}.$$

(152)

Taking into account that $\mathbf{u}={\mathbf{u}}_0+\tilde{\mathbf{u}}$, $\mathbf{H}={\mathbf{H}}_0+\tilde{\mathbf{H}}$, from (151), (152) we deduce, using the estimates in (127) for $\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}$ and $\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}$ that

$${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\le ({\mathcal{R}}_1+1)C_{{\epsilon }_0}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+\sqrt{\kappa }{\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s{\parallel q\parallel }_{s,\mathsf{\Gamma }}+$$

$$+\sqrt{12}\frac{C_0}{{\alpha }_1}\sqrt{\frac{\kappa }{\mathcal{P}m}}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\frac{\sqrt{24}{\beta }_{1}d}{{\alpha }_0\nu }=$$

$$=a_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+a_2\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+a_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+a_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+a_5\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }},$$

(153)

$${\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le \sqrt{\frac{\mathcal{P}m}{\kappa }}{\mathcal{R}}_1{C}_{{\epsilon }_0}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+C_{\mathsf{\Gamma }}^s(\sqrt{\mathcal{P}m}{\mathcal{R}}_2+1){\parallel q\parallel }_{s,\mathsf{\Gamma }}+$$

$$+\sqrt{12}\frac{C_0}{{\alpha }_1}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\frac{\sqrt{24}{\beta }_{1}d}{\sqrt{\kappa {\alpha }_0\nu {\alpha }_1{\nu }_m}}=$$

$$={\left(\sqrt{\kappa }\right)}^{-1}\left[b_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+b_2\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+b_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+b_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+b_5\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}\right].$$

(154)

Here, $a_1,a_2,a_3,a_4,a_5$ and $b_1,b_2,b_3,b_4,b_5$ are dimensional, in general case, constants defined by formulas:

$$a_1=\left[({\mathcal{R}}_1+1)C_{{\epsilon }_0}+\frac{\sqrt{24}{\beta }_1}{{\alpha }_0\nu }{d}_1\right],a_2={\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s,$$

$$a_3=\frac{\sqrt{24}{\beta }_1}{{\alpha }_0\nu }{d}_3,a_4=\frac{\sqrt{24}{\beta }_1}{{\alpha }_0\nu }{d}_4,a_5=\sqrt{12}\frac{C_0}{{\alpha }_1}\sqrt{\frac{1}{\mathcal{P}m}},$$

(155)

$$b_1=\left[\sqrt{\frac{\mathcal{P}m}{\kappa }}{\mathcal{R}}_1{C}_{{\epsilon }_0}+\frac{\sqrt{24}{\beta }_1}{\sqrt{\kappa {\alpha }_0\nu {\alpha }_1{\nu }_m}}{d}_1\right],b_2=C_{\mathsf{\Gamma }}^s(\sqrt{\mathcal{P}m}{\mathcal{R}}_2+1),$$

$$b_3=\frac{\sqrt{24}{\beta }_1}{\sqrt{\kappa {\alpha }_0\nu {\alpha }_1{\nu }_m}}{d}_3,b_4=\frac{\sqrt{24}{\beta }_1}{\sqrt{\kappa {\alpha }_0\nu {\alpha }_1{\nu }_m}}{d}_4,b_5=\sqrt{12}\frac{C_0}{{\alpha }_1}.$$

(156)

It is easy to check that each summand in the right-hand side of (153) has the same dimension equal to $L_0^{3/2}/T_0$, coinciding with the dimension of the left-hand side of (153). Similarly, each summand in the right-hand side of (154) has the same dimension equal to $I_0/L_0^{1/2}$, coinciding with the dimension of the left-hand side of (154). Moreover, a simple analysis shows that quantities $a_1$, $a_2$, $a_5$ defined in (155) and quantities $b_1$, $b_2$, $b_5$ defined in (156) are dimensionless and, furthermore, the following quantities are dimensionless:

$$\tilde{a}=\frac{\sqrt{24}{\beta }_1{d}_2}{{\alpha }_0\nu },\tilde{b}=\frac{\sqrt{24}{\beta }_1{d}_2}{\sqrt{{\alpha }_0\nu {\alpha }_1{\nu }_m}},$$

(157)

i.e., $\left[\tilde{a}\right]=1$, $\left[\tilde{b}\right]=1$.

As for coefficients $a_3$, $a_4$ and $b_3$, $b_4$ defined in (155) and (156), they can be represented as

$$a_3=\tilde{a}(d_3/d_2),a_4=\tilde{a}(d_4/d_2),b_3=\tilde{b}(d_3/d_2),b_4=\tilde{b}(d_4/d_2).$$

(158)

Substituting (158) into (153) and (154) and estimating from above the dimensionless multipliers in the obtained expressions, we arrive at the following final estimates for ${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}$ and ${\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}$:

$${\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}\le C_{\mathbf{u}}\left[{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\right.$$

$$\left.+(d_3/d_2){\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+(d_4/d_2){\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}\right],$$

(159)

$${\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le \frac{C_{\mathbf{H}}}{\sqrt{\kappa }}\left[{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+\right.$$

$$\left.+(d_3/d_2){\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+(d_4/d_2){\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}\right].$$

(160)

Here

$$C_{\mathbf{u}}=max(a_1,a_2,\tilde{a},a_5),C_{\mathbf{H}}=max(b_1,b_2,\tilde{b},b_5).$$

(161)

We conclude this Section by deriving estimates similar to (159) and (160) for differences $T=T_1-T_2$ and $p=p_1-p_2$. To derive the estimate for ${\parallel T\parallel }_{1,\mathsf{\Omega }}$, we will use the estimate (131) for $\parallel T_0+\tilde{T}{\parallel }_{1,\mathsf{\Omega }}$ containing $\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}$, and the estimate (151) for $\parallel \tilde{\mathbf{u}}{\parallel }_{1,\mathsf{\Omega }}$. Using (131), (151) and the estimates in (127) for $\parallel {\mathbf{u}}_0{\parallel }_{1,\mathsf{\Omega }}$ and $\parallel {\mathbf{H}}_0{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}$, we derive

$${\parallel T\parallel }_{1,\mathsf{\Omega }}\le d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+d_2{\parallel \tilde{\mathbf{u}}\parallel }_{1,\mathsf{\Omega }}\le$$

$$\le d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+$$

$$+d_2[{\mathcal{R}}_1{C}_{{\epsilon }_0}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}]+d_2[\sqrt{\kappa }{\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s{\parallel q\parallel }_{s,\mathsf{\Gamma }}]+\sqrt{12}{d}_2\frac{C_0}{{\alpha }_1}\sqrt{\frac{\kappa }{\mathcal{P}m}}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+$$

$$+\frac{\sqrt{24}{\beta }_1{d}_2}{{\alpha }_0\nu }[d_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+d_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}]=$$

$$=\left[d_1+d_2{\mathcal{R}}_1{C}_{{\epsilon }_0}+\frac{\sqrt{24}{\beta }_1{d}_2{d}_1}{{\alpha }_0\nu }\right]{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+$$

$$+d_3\left[1+\frac{\sqrt{24}{\beta }_1{d}_2}{{\alpha }_0\nu }\right]{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+d_2\sqrt{\kappa }{\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s{\parallel q\parallel }_{s,\mathsf{\Gamma }}+$$

$$+d_4\left[1+\frac{\sqrt{24}{\beta }_1{d}_2}{{\alpha }_0\nu }\right]{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+\sqrt{12}{d}_2\frac{C_0}{{\alpha }_1}\sqrt{\frac{\kappa }{\mathcal{P}m}}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}=$$

$$=c_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+c_2\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+$$

$$+c_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+c_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+c_5\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}.$$

(162)

Here

$$c_1=d_1(1+{\mathcal{R}}_1+\tilde{a}),c_2=d_2{\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s,c_3=d_3(1+\tilde{a}),$$

$$c_4=d_4(1+\tilde{a}),c_5=\sqrt{12}{d}_2\frac{C_0}{{\alpha }_1\sqrt{\mathcal{P}m}}.$$

(163)

Note, that each summand in the right-hand side of (162) has the same dimension equal to $K_0{L}^{1/2}$, coinciding with the dimension of the left-hand side of (162).

Analysis of Formula (163) shows that each of coefficients $c_1$, $c_2$, $c_3$, $c_4,c_5$ in (163) is the product of some dimensionless multiplier by one of the coefficients $d_1$, $d_2$, $d_3$, $d_4$. With this in mind, substituting expressions (163) for coefficients $c_1$, $c_2$, $c_3$, $c_4,c_5$ into (162) and estimating the mentioned dimensionless multipliers from above, we arrive at the following final estimate for ${\parallel T\parallel }_{1,\mathsf{\Omega }}$:

$${\parallel T\parallel }_{1,\mathsf{\Omega }}\le \left(C_T{d}_2\right)\left[{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}+\sqrt{\kappa }{\parallel q\parallel }_{s,\mathsf{\Gamma }}+\right.$$

$$\left.+(d_3/d_2){\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+(d_4/d_2){\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}+\sqrt{\kappa }{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}\right].$$

(164)

Here,

$$C_T=max[(d_1/d_2)(1+{\mathcal{R}}_1+\tilde{a}),{\mathcal{R}}_2{C}_{\mathsf{\Gamma }}^s,(1+\tilde{a}),\sqrt{12}\frac{C_0}{{\alpha }_1\sqrt{\mathcal{P}m}}].$$

(165)

Based on (117), we now derive a similar estimate for the difference $p=p_1-p_2$. In view of inf-sup condition (26) for the function p and for any (small enough) number $\delta >0$, there exists a function ${\mathbf{v}}_0\in H_0^1{\left(\mathsf{\Omega }\right)}^3$, ${\mathbf{v}}_0\ne \mathbf{0}$ such that

$$-(\mathrm{div}{\mathbf{v}}_0,p)\ge {\beta }_0\parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel p\parallel }_{\mathsf{\Omega }},{\beta }_0=(\beta -\delta )>0.$$

(166)

Setting $\mathbf{v}={\mathbf{v}}_0$, $\mathbf{\Psi }=\mathbf{0}$ in (117), we obtain

$$\nu (\nabla \mathbf{u},\nabla {\mathbf{v}}_0)+[((\mathbf{u}·\nabla ){\mathbf{u}}_1,{\mathbf{v}}_0)+(({\mathbf{u}}_2·\nabla )\mathbf{u},{\mathbf{v}}_0)]-\kappa [(\mathrm{rot}{\mathbf{H}}_1\times {\mathbf{H},\mathbf{v}}_0)+$$

$$+(\mathrm{rot}\mathbf{H}\times {\mathbf{H}}_2,{\mathbf{v}}_0)]-(\mathrm{div}{\mathbf{v}}_0,p)+(\mathbf{b}T,{\mathbf{v}}_0)=0.$$

(167)

Using the previous estimate (166) for $-(\mathrm{div}{\mathbf{v}}_0,p)$ and (21), (23)–(25), from (167) we deduce that

$${\beta }_0\parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel p\parallel }_{\mathsf{\Omega }}\le -(\mathrm{div}{\mathbf{v}}_0,p)\le \nu \parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}+$$

$$+2{\gamma }_0{M}_{\mathbf{u}}^0\parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}+2\kappa {\gamma }_1{M}_{\mathbf{H}}^0\parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+{\beta }_1{\parallel T\parallel }_{1,\mathsf{\Omega }}{\parallel {\mathbf{v}}_0\parallel }_{1,\mathsf{\Omega }}.$$

(168)

Dividing (168) by $\parallel {\mathbf{v}}_0{\parallel }_{1,\mathsf{\Omega }}\ne 0$ and using estimates (159), (160) and (164) we obtain

$${\parallel p\parallel }_{\mathsf{\Omega }}\le {\beta }_0^{-1}[(\nu +2{\gamma }_0{M}_{\mathbf{u}}^0){\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}+2\kappa {\gamma }_1{M}_{\mathbf{H}}^0{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+{\beta }_1{\parallel T\parallel }_{1,\mathsf{\Omega }}]=$$

$$={\beta }_0^{-1}{\alpha }_0\nu [({\alpha }_0^{-1}+2\mathcal{R}e){\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}+2\sqrt{\kappa }{\mathcal{H}a\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+\frac{{\beta }_1}{{\alpha }_0\nu }{\parallel T\parallel }_{1,\mathsf{\Omega }}].$$

(169)

Taking into account (159), (160) and (164) we obtain the following estimate:

$${\parallel p\parallel }_{\mathsf{\Omega }}\le C_p\nu \left[(\parallel \mathbf{g}{\parallel }_{1/2,\mathsf{\Gamma }}+\sqrt{\kappa }\parallel q{\parallel }_{s,\mathsf{\Gamma }}+\sqrt{\kappa }\parallel \mathbf{j}{\parallel }_{\mathsf{\Omega }}+\right.$$

$$\left.+(d_3/d_2){\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}+(d_4/d_2){\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}\right].$$

(170)

Here, $C_p$ is a dimensionless constant defined by

$$C_p\equiv {\alpha }_0{\beta }_0^{-1}[({\alpha }_0^{-1}+2\mathcal{R}e)C_{\mathbf{u}}+2\mathcal{H}a{C}_{\mathbf{H}}+\frac{{\beta }_1{d}_2}{{\alpha }_0\nu }{C}_T].$$

(171)

6. Analysis of Uniqueness and Stability for Solutions to Control Problems

In this section, using properties of optimal solutions established in Section 5 and, in particular, the inequality (125) and estimates (159), (160), (164), (170), we will derive stability estimates of optimal solutions for two cost functionals belonging to the set (79) of possible cost functionals. We begin with a consideration of the case when $I=I_1$ in (78), i.e., we consider the control problem

$$J(\mathbf{x},u)\equiv \frac{{\mu }_0}{2}\parallel \mathbf{u}-{\mathbf{u}}_d{\parallel }_Q^2+\frac{{\mu }_1}{2}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\frac{{\mu }_2}{2}{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\frac{{\mu }_3}{2}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\frac{{\mu }_4}{2}{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\to \inf ,$$

$$F(\mathbf{x},u,\mathbf{j})=0,(\mathbf{x},u,\mathbf{j})\in X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right).$$

(172)

Here, as usual, $\mathbf{x}=(\mathbf{u},\mathbf{H},T,p)$, $u=(\mathbf{g},q,\psi ,\chi )$. Denote by $({\mathbf{x}}_1,u_1)\equiv$ (${\mathbf{u}}_1$, ${\mathbf{H}}_1$, $T_1$, $p_1$, ${\mathbf{g}}_1$, $q_1$, ${\psi }_1$, ${\chi }_1$) a solution to problem (172) that corresponds to the pair of functions ${\mathbf{u}}_d\equiv {\mathbf{u}}_d^{\left(1\right)}\in {\mathbf{L}}^2\left(Q\right)$ and $\mathbf{j}={\mathbf{j}}_1\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$. By $({\mathbf{x}}_2,u_2)\equiv$ (${\mathbf{u}}_2$, ${\mathbf{H}}_2$, $T_2$, $p_2$, ${\mathbf{g}}_2$, $q_2$, ${\psi }_2$, ${\chi }_2$), we denote a solution to problem (172) that corresponds to another pair of functions ${\tilde{\mathbf{u}}}_d\equiv {\mathbf{u}}_d^{\left(2\right)}\in {\mathbf{L}}^2\left(Q\right)$ and $\tilde{\mathbf{j}}={\mathbf{j}}_2\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$.

We define dimensionless parameters $\mathcal{R}{e}^0$ (Reynolds number for the data ${\mathbf{u}}_d^{\left(i\right)}$) and ${\mathcal{R}}_3$ by

$$\mathcal{R}{e}^0=({\gamma }_0/{\delta }_0\nu l)max(\parallel {\mathbf{u}}_d^{\left(1\right)}{\parallel }_Q,\parallel {\mathbf{u}}_d^{\left(2\right)}{\parallel }_Q),{\mathcal{R}}_3=\frac{{\beta }_1{d}_2}{{\alpha }_2\lambda },$$

(173)

where l is a dimensional constant defined in Section 2, and define constants a, $\gamma$, b and nonnegative function $\phi :[0,\infty )\to [0,\infty )$ by formulas

$$a=4C_1\gamma ({\nu }_1/\sqrt{\kappa })\sqrt{\mathcal{P}m}(\mathcal{R}e+\mathcal{R}{e}^0),\gamma =l^2{\gamma }_0^{-1},$$

(174)

$$b=20\gamma (\mathcal{R}e+\mathcal{R}{e}^0)({\gamma }_0{C}_{\mathbf{u}}^2+{\gamma }_1{C}_{\mathbf{H}}^2+{\gamma }_1\sqrt{\mathcal{P}m}{C}_{\mathbf{u}}{C}_{\mathbf{H}}+{\gamma }_2{\mathcal{R}}_3{C}_{\mathbf{u}}{C}_T),$$

(175)

$${\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{)=(a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2{)}^{1/2}.$$

(176)

In (175) $C_{\mathbf{u}}$, $C_{\mathbf{H}}$ and $C_T$ are dimensionless constants defined in (161) and (165).

Remark 1.

A simple analysis shows that

$$\left[a\right]=L^{11/2}/\left(T_0^2{I}_0\right),\left[b\right]=L_0^2{,[a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{]=[b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2]=L_0^5/T_0^2{,\left[\phi \right(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{\left)\right]}^2=L_0^5/T_0^2.$$

Thus, the function φ given by (176) is defined correctly in terms of dimensions.

We assume that the data for the problem (172) or parameters ${\mu }_0$, ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ and ${\mu }_4$ are such that the following condition with some sufficiently small $\epsilon >0$ takes place:

$${\mu }_{0}b\le (1-\epsilon ){\mu }_1,{\mu }_{0}b\kappa \le (1-\epsilon ){\mu }_2,$$

$${\mu }_{0}b{(d_3/d_2)}^2\le (1-\epsilon ){\mu }_3,{\mu }_{0}b{(d_4/d_2)}^2\le (1-\epsilon ){\mu }_4,\epsilon \in (0,1).$$

(177)

Lemma 6.

Let, under Hypothesises 1–3 and 8 and (112), (177), a pair $({\mathbf{x}}_i,u_i)=$ (${\mathbf{u}}_i$, ${\mathbf{H}}_i$, $T_i$, $p_i$, ${\mathbf{g}}_i$, $q_i$, ${\psi }_i$, ${\chi }_i$) be a solution to problem (172) corresponding to given pair ${\mathbf{u}}_d^{\left(i\right)}\in {\mathbf{L}}^2\left(Q\right)$ and $\mathbf{j}={\mathbf{j}}_i\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$, $i=1,2$, where $Q\subset \mathsf{\Omega }$ is an arbitrary nonempty open subset. Then the following estimate for $\mathbf{u}\equiv {\mathbf{u}}_1-{\mathbf{u}}_2$ holds:

$$\parallel {\mathbf{u}}_1-{\mathbf{u}}_2{\parallel }_Q\le \mathsf{\Delta }\equiv \parallel {\mathbf{u}}_d^{\left(1\right)}-{\mathbf{u}}_d^{\left(2\right)}{\parallel }_Q+\phi (\parallel {\mathbf{j}}_1-{\mathbf{j}}_2{\parallel }_{\mathsf{\Omega }}).$$

(178)

Proof of Lemma 6.

Setting ${\mathbf{u}}_d={\mathbf{u}}_d^{\left(1\right)}-{\mathbf{u}}_d^{\left(2\right)}$, in addition to (116), we have that ${\left(I_1\right)}_p^{\prime }=0$, ${\left(I_1\right)}_{\mathbf{H}}^{\prime }=0$, ${\left(I_1\right)}_T^{\prime }=0$, but

$$\langle {\left(I_1\right)}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_i\right),\mathbf{w}\rangle =2{({\mathbf{u}}_i-{\mathbf{u}}_d^{\left(i\right)},\mathbf{w})}_Q,\langle {\left(I_1\right)}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_1\right)-{\left({\tilde{I}}_1\right)}_{\mathbf{u}}^{\prime }\left({\mathbf{x}}_2\right),\mathbf{u}\rangle =$$

$$=2{({\mathbf{u}-\mathbf{u}}_d,\mathbf{u})}_Q={2(\parallel \mathbf{u}\parallel }_Q^2-{(\mathbf{u},{\mathbf{u}}_d)}_Q).$$

(179)

In view of (179), identities (113)–(115) for adjoint states ${\mathbf{y}}_i^{*}=({\xi }_i$, ${\eta }_i$, ${\sigma }_i$, ${\zeta }_1^{\left(i\right)}$, ${\zeta }_2^{\left(i\right)}$, ${\theta }_i$, ${\zeta }_3^{\left(i\right)})$, $i=1,2$, corresponding to solutions $({\mathbf{u}}_i$, ${\mathbf{H}}_i$, $T_i$, $p_i$, $u_i)$ and the main inequality (125) for differences $\mathbf{u}$, $\mathbf{H}$, T, p, $\mathbf{g}$, q, $\psi$, $\chi$, $\mathbf{j}$ defined in (116), take the form

$$\nu (\nabla \mathbf{w},\nabla {\xi }_i)+{\nu }_1(\mathrm{rot}\mathbf{h},\mathrm{rot}{\eta }_i)+(({\mathbf{u}}_i·\nabla )\mathbf{w},{\xi }_i)+$$

$$+((\mathbf{w}·\nabla ){\mathbf{u}}_i,{\xi }_i)+\kappa [(\mathrm{rot}{\eta }_i\times {\mathbf{H}}_i,\mathbf{w})+$$

$$+(\mathrm{rot}{\eta }_i\times \mathbf{h},{\mathbf{u}}_i)]-\kappa [(\mathrm{rot}{\mathbf{H}}_i\times \mathbf{h},{\xi }_i)+(\mathrm{rot}\mathbf{h}\times {\mathbf{H}}_i,{\xi }_i)]+$$

$$+\tilde{\kappa }(\mathbf{w}·\nabla T_i,{\theta }_i)-(\mathrm{div}\mathbf{w},{\sigma }_i)+{\langle {\zeta }_1^{\left(i\right)},\mathbf{w}\rangle }_{\mathsf{\Gamma }}+{\langle {\zeta }_2^{\left(i\right)},\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}=$$

$$=-{\mu }_0{({\mathbf{u}}_i-{\mathbf{u}}_d^{\left(i\right)},\mathbf{w})}_Q\forall \left(\mathbf{w},\mathbf{h}\right)\in H_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right);{\xi }_i\in V,i=1,2,$$

(180)

$$\tilde{\kappa }[\lambda \tilde{a}(\tau ,{\theta }_i)+\lambda {(\alpha \tau ,{\theta }_i)}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_i·\nabla \tau ,{\theta }_i)+{\langle {\zeta }_3^{\left(i\right)},\tau \rangle }_{{\mathsf{\Gamma }}_D}]+b_1(\tau ,{\xi }_i)=0\forall \tau \in H^1\left(\mathsf{\Omega }\right),$$

(181)

$$((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)+\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})+$$

$$+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)+{\mu }_0{(\parallel \mathbf{u}\parallel }_Q^2-{(\mathbf{u},{\mathbf{u}}_d)}_Q)\le$$

$$\le -{\nu }_1(\mathbf{j},\mathrm{rot}\eta )-{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-{\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2.$$

(182)

Firstly, we estimate adjoint state variables ${\xi }_i$, ${\eta }_i$ and ${\theta }_i$ via $(\mathcal{R}e+\mathcal{R}{e}^0)$. To this end, we set $\mathbf{w}={\xi }_i$, $\mathbf{h}={\eta }_i$ in (180) and $\tau ={\theta }_i$ in (181). Using (18) and conditions $\mathrm{div}{\xi }_i=0$ in $\mathsf{\Omega }$, ${\xi }_i{|}_{\mathsf{\Gamma }}=\mathbf{0}$, ${\eta }_i{·\mathbf{n}|}_{\mathsf{\Gamma }}=0$, ${\theta }_i{|}_{{\mathsf{\Gamma }}_D}=0$ we obtain

$$\nu (\nabla {\xi }_i,\nabla {\xi }_i)+{\nu }_1(\mathrm{rot}{\eta }_i,\mathrm{rot}{\eta }_i)+(({\xi }_i·\nabla ){\mathbf{u}}_i,{\xi }_i)+\kappa (\mathrm{rot}{\eta }_i\times {\eta }_i,{\mathbf{u}}_i)-$$

$$-\kappa (\mathrm{rot}{\mathbf{H}}_i\times {\eta }_i,{\xi }_i)+\tilde{\kappa }({\xi }_i·\nabla T_i,{\theta }_i)=-{\mu }_0{({\mathbf{u}}_i-{\mathbf{u}}_d^{\left(i\right)},{\xi }_i)}_Q,$$

(183)

$$\tilde{\kappa }[\lambda \tilde{a}({\theta }_i,{\theta }_i)+\lambda {(\alpha {\theta }_i,{\theta }_i)}_{{\mathsf{\Gamma }}_N}]+b_1({\theta }_i,{\xi }_i)=0.$$

(184)

Taking into account estimates (21)–(25), (109) and (111), (37), (173), we have

$$(\nabla {\xi }_i,\nabla {\xi }_i)\ge {\delta }_0{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}^2,$$

$$|(({\xi }_i·\nabla ){\mathbf{u}}_i,{\xi }_i)|\le {\gamma }_0\parallel {\mathbf{u}}_i{\parallel }_{1,\mathsf{\Omega }}\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}^2\le {\gamma }_0{M}_{\mathbf{u}}^0{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}^2,$$

(185)

$$(\mathrm{rot}{\eta }_i,\mathrm{rot}{\eta }_i)\ge {\delta }_1\parallel {\eta }_i{\parallel }_{1,\mathsf{\Omega }}^2,|(\mathrm{rot}{\eta }_i\times {\eta }_i,{\mathbf{u}}_i)|\le {\gamma }_1{M}_{\mathbf{u}}^0{\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}^2,$$

(186)

$$\kappa |(\mathrm{rot}{\mathbf{H}}_i\times {\eta }_i,{\xi }_i)|\le {\gamma }_1\kappa M_{\mathbf{H}}^0\parallel {\eta }_i{\parallel }_{1,\mathsf{\Omega }}{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}\le$$

$$\le {\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0(\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}^2+\kappa \parallel {\eta }_i{\parallel }_{1,\mathsf{\Omega }}^2),$$

(187)

$$\parallel {\mathbf{u}}_i-{\mathbf{u}}_d^{\left(i\right)}{\parallel }_Q\le \parallel {\mathbf{u}}_i{\parallel }_Q+\parallel {\mathbf{u}}_d^{\left(i\right)}{\parallel }_Q\le l{M}_{\mathbf{u}}^0+{\parallel {\mathbf{u}}_d^{\left(i\right)}\parallel }_Q\le {\delta }_0\nu l{\gamma }_0^{-1}(\mathcal{R}e+\mathcal{R}{e}^0).$$

(188)

Using (23) and (28) we derive successfully from (184) that

$$\parallel {\theta }_i{\parallel }_{1,\mathsf{\Omega }}\le ({\beta }_1/{\alpha }_2\lambda \tilde{\kappa }){\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }},$$

(189)

$$|\tilde{\kappa }({\xi }_i·\nabla T_i,{\theta }_i)|\le {\gamma }_2\tilde{\kappa }{M}_T^0\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}\parallel {\theta }_i{\parallel }_{1,\mathsf{\Omega }}\le \frac{{\beta }_1{\gamma }_2}{{\alpha }_2\lambda }{M}_T^0{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}^2.$$

(190)

In virtue of (185)–(188) and (145), we infer from (183) that

$$({\delta }_0\nu /2)\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}^2+({\delta }_1{\nu }_m/2)\kappa {\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$\le [{\delta }_0\nu -{\gamma }_0{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0+\frac{{\beta }_1{\gamma }_2}{{\alpha }_2\lambda }{M}_T^0]{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+[{\delta }_1{\nu }_m-{\gamma }_1{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0]\kappa {\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$\le {\mu }_{0}l(l{M}_{\mathbf{u}}^0+\parallel {\mathbf{u}}_d^{\left(i\right)}{\parallel }_Q)\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}\le {\mu }_0{\delta }_0\nu \gamma (\mathcal{R}e+\mathcal{R}{e}^0){\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }},$$

(191)

where $\gamma =l^2{\gamma }_0^{-1}$. Using (111), (185) and (186) we conclude from (191) that

$$\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}\le 2{\mu }_0\gamma (\mathcal{R}e+\mathcal{R}{e}^0),$$

$$\sqrt{\kappa }{\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}\le 2{\mu }_0\gamma \sqrt{\mathcal{P}m}(\mathcal{R}e+\mathcal{R}{e}^0),i=1,2,$$

(192)

$$\tilde{\kappa }{\parallel {\theta }_i\parallel }_{1,\mathsf{\Omega }}\le ({\beta }_1/{\alpha }_2\lambda )2{\mu }_0\gamma (\mathcal{R}e+\mathcal{R}{e}^0).$$

(193)

Besides, from (22), (174) and (192) it follows that

$$|{\nu }_1(\mathbf{j},\mathrm{rot}\eta )|\le C_1{\nu }_1{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}(4/\sqrt{\kappa })\gamma \sqrt{\mathcal{P}m}(\mathcal{R}e+\mathcal{R}{e}^0)=a{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}.$$

(194)

Using (19)–(25), (159), (160), (164), (177) and (192), (193), we have

$$\left|((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)+\right.$$

$$\left.+\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)\right|\le$$

$$\le {\gamma }_0{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2(\parallel {\xi }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\xi }_2{\parallel }_{1,\mathsf{\Omega }})+{\gamma }_1{\kappa \parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2(\parallel {\xi }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\xi }_2{\parallel }_{1,\mathsf{\Omega }})+$$

$$+{\gamma }_1{\kappa \parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}(\parallel {\eta }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\eta }_2{\parallel }_{1,\mathsf{\Omega }})+$$

$$+{\gamma }_2\tilde{\kappa }{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel T\parallel }_{1,\mathsf{\Omega }}(\parallel {\theta }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\theta }_2{\parallel }_{1,\mathsf{\Omega }})\le$$

$$\le 20{\mu }_0\gamma (\mathcal{R}e+\mathcal{R}{e}^0)\left[{\gamma }_0{C}_{\mathbf{u}}^2+{\gamma }_1{C}_{\mathbf{H}}^2+{\gamma }_1\sqrt{\mathcal{P}m}{C}_{\mathbf{u}}{C}_{\mathbf{H}}+\right.$$

$$\left.+{\gamma }_2({\beta }_1{d}_2/{\alpha }_2\lambda )C_{\mathbf{u}}{C}_T\right]\left[{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\right.$$

$$\left.+{\kappa \parallel q\parallel }_{s,\mathsf{\Gamma }}^2+{\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+{(d_3/d_2)}^2{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+{(d_4/d_2)}^2{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\right]=$$

$$={\mu }_0{b[\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+{\kappa \parallel q\parallel }_{s,\mathsf{\Gamma }}^2+{\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+{(d_3/d_2)}^2{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+{(d_4/d_2)}^2{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2]\le$$

$$\le (1-\epsilon ){\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+(1-\epsilon ){\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+$$

$$+(1-\epsilon ){\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+(1-\epsilon ){\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2+{\mu }_0{a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{\mu }_{0}b\kappa {\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2.$$

(195)

Taking into account (194), (195) from (182) we arrive at

$${\mu }_0{(\parallel \mathbf{u}\parallel }_Q^2-{\left({\mathbf{u},\mathbf{u}}_d\right)}_Q)\le -((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)+\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)-$$

$$-\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})-\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)-{\nu }_1(\mathbf{j},\mathrm{rot}\eta )-{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-$$

$$-{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-{\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\le$$

$$\le -\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2+{\mu }_0{a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{\mu }_{0}b\kappa {\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2.$$

(196)

Omitting the nonpositive term $-\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2$, we derive from (196) that

$${\parallel \mathbf{u}\parallel }_Q^2-{\parallel \mathbf{u}\parallel }_Q\parallel {\mathbf{u}}_d{\parallel }_Q\le {a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2\equiv {\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}).$$

(197)

Solving the quadratic with respect to ${\parallel \mathbf{u}\parallel }_Q$ inequality (197) we have

$${\parallel \mathbf{u}\parallel }_Q\le \parallel {\mathbf{u}}_d{\parallel }_Q+{\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}).$$

(198)

Since $\mathbf{u}={\mathbf{u}}_1-{\mathbf{u}}_2$, ${\mathbf{u}}_d={\mathbf{u}}_d^{\left(1\right)}-{\mathbf{u}}_d^{\left(2\right)}$, $\mathbf{j}={\mathbf{j}}_1-{\mathbf{j}}_2$, Lemma 6 is proved. □

If $Q=\mathsf{\Omega }$ and ${\mathbf{j}}_1={\mathbf{j}}_2$ the estimate (178) has the sense of stability estimate in ${\mathbf{L}}^2\left(\mathsf{\Omega }\right)$ norm for the component $\widehat{\mathbf{u}}$ of the solution $(\widehat{\mathbf{u}}$, $\widehat{\mathbf{H}}$, $\widehat{p}$, $\widehat{T}$, $\widehat{\mathbf{g}}$, $\widehat{q}$, $\widehat{\psi }$, $\widehat{\chi })$ of problem (172) with respect to small disturbances of function ${\mathbf{u}}_d\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$ in the norm of ${\mathbf{L}}^2\left(\mathsf{\Omega }\right)$. Additionally, if ${\mathbf{u}}_d^{\left(1\right)}={\mathbf{u}}_d^{\left(2\right)}$ and ${\mathbf{j}}_1={\mathbf{j}}_2$ we conclude from (178) that ${\mathbf{u}}_1={\mathbf{u}}_2$ in Q. This together with (196) yields that ${\mathbf{g}}_1={\mathbf{g}}_2$, $q_1=q_2$, ${\psi }_1={\psi }_2$, ${\chi }_1={\chi }_2$. In turn, it follows from this fact and (159), (160), (164) and (170) that ${\mathbf{u}}_1={\mathbf{u}}_2$, ${\mathbf{H}}_1={\mathbf{H}}_2$, $T_1=T_2$ and $p_1=p_2$ in $\mathsf{\Omega }$. The latter is equivalent to the uniqueness of the solution of (172).

Using (178) we are now able to obtain stability estimates for all differences $\mathbf{u},\mathbf{H},T$ and p even in situation where $Q\subset \mathsf{\Omega }$, i.e., Q is only a part of $\mathsf{\Omega }$. To this end, we consider inequality (196). By (197), (198) we deduce from (196) that

$$\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\le$$

$$\le {\mu }_0{(-\parallel \mathbf{u}\parallel }_Q^2+{\parallel \mathbf{u}\parallel }_Q\parallel {\mathbf{u}}_d{\parallel }_Q)+{\mu }_0{\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})\le {\mu }_0\parallel {\mathbf{u}}_d{\parallel }_Q^2+{\mu }_0{\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})\le {\mu }_0{\mathsf{\Delta }}^2,$$

(199)

where $\mathsf{\Delta }$ is defined in (178). From (159), (160), (164), (170) and (199), we arrive at

$$\parallel {\mathbf{g}}_1-{\mathbf{g}}_2{\parallel }_{1/2,\mathsf{\Gamma }}\le \sqrt{{\mu }_0/\left(\epsilon {\mu }_1\right)}\mathsf{\Delta },{\parallel q_1-q_2\parallel }_{s,\mathsf{\Gamma }}\le \sqrt{{\mu }_0/\left(\epsilon {\mu }_2\right)}\mathsf{\Delta },$$

(200)

$$\parallel {\psi }_1-{\psi }_2{\parallel }_{1/2,{\mathsf{\Gamma }}_D}\le \sqrt{{\mu }_0/\left(\epsilon {\mu }_3\right)}\mathsf{\Delta },{\parallel {\chi }_1-{\chi }_2\parallel }_{{\mathsf{\Gamma }}_N}\le \sqrt{{\mu }_0/\left(\epsilon {\mu }_4\right)}\mathsf{\Delta },$$

(201)

$$\parallel {\mathbf{u}}_1-{\mathbf{u}}_2{\parallel }_{1,\mathsf{\Omega }}\le C_{\mathbf{u}}\left[\sqrt{{\mu }_0/\left(\epsilon {\mu }_1\right)}+\sqrt{\kappa }\sqrt{{\mu }_0/\left(\epsilon {\mu }_2\right)}+(d_3/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_3\right)}+\right.$$

$$\left.+(d_4/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_4\right)}\right]\mathsf{\Delta }+C_{\mathbf{u}}\sqrt{\kappa }{\parallel {\mathbf{j}}_1-{\mathbf{j}}_2\parallel }_{\mathsf{\Omega }},$$

(202)

$$\parallel {\mathbf{H}}_1-{\mathbf{H}}_2{\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}\le \frac{C_{\mathbf{H}}}{\sqrt{\kappa }}\left[\sqrt{{\mu }_0/\left(\epsilon {\mu }_1\right)}+\sqrt{\kappa }\sqrt{{\mu }_0/\left(\epsilon {\mu }_2\right)}+(d_3/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_3\right)}+\right.$$

$$\left.+(d_4/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_4\right)}\right]\mathsf{\Delta }+C_{\mathbf{H}}{\parallel {\mathbf{j}}_1-{\mathbf{j}}_2\parallel }_{\mathsf{\Omega }},$$

(203)

$$\parallel T_1-T_2{\parallel }_{1,\mathsf{\Omega }}\le C_T{d}_2\left[\sqrt{{\mu }_0/\left(\epsilon {\mu }_1\right)}+\sqrt{\kappa }\sqrt{{\mu }_0/\left(\epsilon {\mu }_2\right)}+(d_3/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_3\right)}+\right.$$

$$\left.+(d_4/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_4\right)}\right]\mathsf{\Delta }+C_T{d}_2\sqrt{\kappa }{\parallel {\mathbf{j}}_1-{\mathbf{j}}_2\parallel }_{\mathsf{\Omega }},$$

(204)

$$\parallel p_1-p_2{\parallel }_{\mathsf{\Omega }}\le C_p\nu \left[\sqrt{{\mu }_0/\left(\epsilon {\mu }_1\right)}+\sqrt{\kappa }\sqrt{{\mu }_0/\left(\epsilon {\mu }_2\right)}+(d_3/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_3\right)}+\right.$$

$$\left.+(d_4/d_2)\sqrt{{\mu }_0/\left(\epsilon {\mu }_4\right)}\right]\mathsf{\Delta }+C_p\nu \sqrt{\kappa }{\parallel {\mathbf{j}}_1-{\mathbf{j}}_2\parallel }_{\mathsf{\Omega }}.$$

(205)

Let us describe the obtained result as the following theorem:

Theorem 8.

Let parameters $\mathcal{R}{e}^0$, ${\mathcal{R}}_3$, a, γ, b and function $\phi :[0,\infty )\to [0,\infty )$ be defined by (173)–(176) and let assumptions of Lemma 6 be fulfilled. Then, the stability estimates (200)–(205) for problem (172) hold where $\mathsf{\Delta }=\parallel {\mathbf{u}}_d^{\left(1\right)}-{\mathbf{u}}_d^{\left(2\right)}{\parallel }_Q+\phi (\parallel {\mathbf{j}}_1-{\mathbf{j}}_2{\parallel }_{\mathsf{\Omega }})$.

Now, we consider the case when $I=I_3$ in (78), i.e., we consider the control problem

$$J(\mathbf{x},u)\equiv \frac{{\mu }_0}{2}\parallel T-T_d{\parallel }_Q^2+\frac{{\mu }_1}{2}{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\frac{{\mu }_2}{2}{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\frac{{\mu }_3}{2}{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\frac{{\mu }_4}{2}{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\to \inf ,$$

$$F(\mathbf{x},u,\mathbf{j})=0,(\mathbf{x},u,\mathbf{j})\in X\times K\times {\mathbf{L}}^2\left(\mathsf{\Omega }\right).$$

(206)

Again we denote by $({\mathbf{x}}_1,u_1)\equiv$ (${\mathbf{u}}_1$, ${\mathbf{H}}_1$, $T_1$, $p_1$, ${\mathbf{g}}_1$, $q_1$, ${\psi }_1$, ${\chi }_1$) a solution to problem (206) that corresponds to the pair $T_d\equiv T_d^{\left(1\right)}\in L^2\left(Q\right)$ and $\mathbf{j}={\mathbf{j}}_1\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$. By $({\mathbf{x}}_2,u_2)\equiv$ (${\mathbf{u}}_2$, ${\mathbf{H}}_2$, $T_2$, $p_2$, ${\mathbf{g}}_2$, $q_2$, ${\psi }_2$, ${\chi }_2$), we denote a solution to problem (206) that corresponds to another pair ${\tilde{T}}_d\equiv T_d^{\left(2\right)}\in L^2\left(Q\right)$ and $\mathbf{j}={\mathbf{j}}_2\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$.

We define constants ${\tilde{M}}_T^0$, ${\mathcal{R}}_3$, a, b and nonnegative function $\phi :[0,\infty )\to [0,\infty )$ by

$${\tilde{M}}_T^0=l^2{M}_T^0+lmax(\parallel T_d^{\left(1\right)}{\parallel }_Q,\parallel T_d^{\left(2\right)}{\parallel }_Q),{\mathcal{R}}_3=\frac{{\beta }_1{d}_2}{{\alpha }_2\lambda },a=\frac{4C_1\sqrt{\kappa }{\nu }_m\sqrt{\mathcal{P}m}\mathcal{R}a{\tilde{M}}_T^0}{{\beta }_1},$$

(207)

$$b=\frac{20{\tilde{M}}_T^0}{{\beta }_1}\left[{\gamma }_0{C}_{\mathbf{u}}^2\mathcal{R}a+{\gamma }_1{C}_{\mathbf{H}}^2\mathcal{R}a+{\gamma }_1{C}_{\mathbf{u}}{C}_{\mathbf{H}}\sqrt{\mathcal{P}m}\mathcal{R}a+{\gamma }_2{C}_{\mathbf{u}}{C}_T{\mathcal{R}}_3(\mathcal{R}a+0.5)\right],$$

(208)

$${\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{)=(a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2{)}^{1/2}.$$

(209)

Here, $C_{\mathbf{u}}$, $C_{\mathbf{H}}$, $C_T$ are dimensionless constants defined in (161), (165).

Remark 2.

A simple analysis shows that

$$\left[a\right]=K_0^2{L}_0^{7/2}/I_0,\left[b\right]=K_0^2{T}_0^2{,[a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{]=[b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2]=K_0^2{L}_0^3{,\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}{)}^2=K_0^2{L}_0^3.$$

This means that the function φ specified in (209) is defined correctly in terms of dimensions.

We assume that the data for the problem (206) or parameters ${\mu }_0$, ${\mu }_1$, ${\mu }_2$, ${\mu }_3$ and ${\mu }_4$ are such that the following condition with some sufficiently small $\epsilon >0$ takes place:

$${\mu }_{0}b\le (1-\epsilon ){\mu }_1,{\mu }_{0}b\kappa \le (1-\epsilon ){\mu }_2,$$

$${\mu }_{0}b{(d_3/d_2)}^2\le (1-\epsilon ){\mu }_3,{\mu }_{0}b{(d_4/d_2)}^2\le (1-\epsilon ){\mu }_4,\epsilon \in (0,1).$$

(210)

Lemma 7.

Let, under Hypothesises 1–3 and 8 and (112), (210), a pair $({\mathbf{x}}_i,u_i)=$ (${\mathbf{u}}_i$, ${\mathbf{H}}_i$, $T_i$, $p_i$, ${\mathbf{g}}_i$, $q_i$, ${\psi }_i$, ${\chi }_i$) be a solution to the problem (206) corresponding to given pair $T_d^{\left(i\right)}\in L^2\left(Q\right)$ and $\mathbf{j}={\mathbf{j}}_i\in {\mathbf{L}}^2\left(\mathsf{\Omega }\right)$, $i=1,2$, where $Q\subset \mathsf{\Omega }$ is an arbitrary nonempty open subset. Then, the following estimate for $T\equiv T_1-T_2$ holds:

$$\parallel T_1-T_2{\parallel }_Q\le \mathsf{\Delta }\equiv \parallel T_d^{\left(1\right)}-T_d^{\left(2\right)}{\parallel }_Q+\phi (\parallel {\mathbf{j}}_1-{\mathbf{j}}_2{\parallel }_{\mathsf{\Omega }}).$$

(211)

Proof of Lemma 7.

Setting $T_d=T_d^{\left(1\right)}-T_d^{\left(2\right)}$, in addition to (116), we have that ${\left(I_3\right)}_{\mathbf{u}}^{\prime }=0$, ${\left(I_3\right)}_{\mathbf{H}}^{\prime }=0$, ${\left(I_3\right)}_p^{\prime }=0$, but

$$\langle {\left(I_3\right)}_T^{\prime }\left({\mathbf{x}}_i\right),S\rangle =2{(T_i-T_d^{\left(i\right)},S)}_Q,\langle {\left(I_3\right)}_T^{\prime }\left({\mathbf{x}}_1\right)-{\left({\tilde{I}}_3\right)}_T^{\prime }\left({\mathbf{x}}_2\right),T\rangle =$$

$$=2{(T-T_d,T)}_Q={2(\parallel T\parallel }_Q^2-{(T,T_d)}_Q).$$

(212)

In view of (212), identities (113)–(115) for adjoint states ${\mathbf{y}}_i^{*}=({\xi }_i,{\eta }_i,{\sigma }_i,{\zeta }_1^{\left(i\right)},{\zeta }_2^{\left(i\right)},{\theta }_i,{\zeta }_3^{\left(i\right)})$, $i=1,2$, corresponding to solutions $({\mathbf{u}}_i,{\mathbf{H}}_i,T_i,p_i,u_i)$, and the main inequality (125) for differences $\mathbf{u}$, $\mathbf{H}$, T, p, $\mathbf{g}$, q, $\psi$, $\chi$, $\mathbf{j}$ defined in (116), take the form

$$\nu (\nabla \mathbf{w},\nabla {\xi }_i)+{\nu }_1(\mathrm{rot}\mathbf{h},\mathrm{rot}{\eta }_i)+(({\mathbf{u}}_i·\nabla )\mathbf{w},{\xi }_i)+$$

$$+((\mathbf{w}·\nabla ){\mathbf{u}}_i,{\xi }_i)+\kappa [(\mathrm{rot}{\eta }_i\times {\mathbf{H}}_i,\mathbf{w})+$$

$$+(\mathrm{rot}{\eta }_i\times \mathbf{h},{\mathbf{u}}_i)]-\kappa [(\mathrm{rot}{\mathbf{H}}_i\times \mathbf{h},{\xi }_i)+(\mathrm{rot}\mathbf{h}\times {\mathbf{H}}_i,{\xi }_i)]+$$

$$+\tilde{\kappa }(\mathbf{w}·\nabla T_i,{\theta }_i)-(\mathrm{div}\mathbf{w},{\sigma }_i)+{\langle {\zeta }_1^{\left(i\right)},\mathbf{w}\rangle }_{\mathsf{\Gamma }}+{\langle {\zeta }_2^{\left(i\right)},\mathbf{h}·\mathbf{n}\rangle }_{\mathsf{\Gamma }}=$$

$$=0\forall \left(\mathbf{w},\mathbf{h}\right)\in H_T^1\left(\mathsf{\Omega }\right)\times {\mathcal{H}}_m^r\left(\mathsf{\Omega }\right);{\xi }_i\in V,i=1,2,$$

(213)

$$\tilde{\kappa }[\lambda \tilde{a}(\tau ,{\theta }_i)+\lambda {(\alpha \tau ,{\theta }_i)}_{{\mathsf{\Gamma }}_N}+({\mathbf{u}}_i·\nabla \tau ,{\theta }_i)+{\langle {\zeta }_3^{\left(i\right)},\tau \rangle }_{{\mathsf{\Gamma }}_D}]+b_1(\tau ,{\xi }_i)=$$

$$=-{\mu }_0(T_i-T_d^{\left(i\right)},\tau )\forall \tau \in H^1\left(\mathsf{\Omega }\right),$$

(214)

$$((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)+\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})+$$

$$+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)+{\mu }_0{(\parallel T\parallel }_Q^2-{(T,T_d)}_Q)\le$$

$$\le -{\nu }_1(\mathbf{j},\mathrm{rot}\eta )-{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-{\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2.$$

(215)

Firstly, we estimate adjoint state variables ${\xi }_i$, ${\eta }_i$ and ${\theta }_i$ via ${\tilde{M}}_T^0$. To this end, we set $\mathbf{w}={\xi }_i$, $\mathbf{h}={\eta }_i$ in (213) and $\tau ={\theta }_i$ in (214). Using (18) and conditions $\mathrm{div}{\xi }_i=0$ in $\mathsf{\Omega }$, ${\xi }_i{|}_{\mathsf{\Gamma }}=\mathbf{0}$, ${\eta }_i{·\mathbf{n}|}_{\mathsf{\Gamma }}=0$, ${\theta }_i{|}_{{\mathsf{\Gamma }}_D}=0$ we obtain

$$\nu (\nabla {\xi }_i,\nabla {\xi }_i)+{\nu }_1(\mathrm{rot}{\eta }_i,\mathrm{rot}{\eta }_i)+(({\xi }_i·\nabla ){\mathbf{u}}_i,{\xi }_i)+\kappa (\mathrm{rot}{\eta }_i\times {\eta }_i,{\mathbf{u}}_i)-$$

$$-\kappa (\mathrm{rot}{\mathbf{H}}_i\times {\eta }_i,{\xi }_i)+\tilde{\kappa }({\xi }_i·\nabla T_i,{\theta }_i)=0,$$

(216)

$$\tilde{\kappa }[\lambda \tilde{a}({\theta }_i,{\theta }_i)+\lambda {(\alpha {\theta }_i,{\theta }_i)}_{{\mathsf{\Gamma }}_N}]+b_1({\theta }_i,{\xi }_i)=-{\mu }_0(T_i-T_d^{\left(i\right)},{\theta }_i).$$

(217)

Using (37) we easily derive that

$$\parallel T_i-T_d^{\left(i\right)}{\parallel }_Q\le \parallel T_i{\parallel }_Q+\parallel T_d^{\left(i\right)}{\parallel }_Q\le l\parallel T_i{\parallel }_Q+\parallel T_d^{\left(i\right)}{\parallel }_Q\le l{M}_T^0+{\parallel T_d^{\left(i\right)}\parallel }_Q\le {\tilde{M}}_T^0,$$

$$|{(T_i-T_d^{\left(i\right)},{\theta }_i)}_Q|\le \parallel T_i-T_d^{\left(i\right)}{\parallel }_Q\parallel {\theta }_i{\parallel }_Q\le {\tilde{M}}_T^0{\parallel {\theta }_i\parallel }_{1,\mathsf{\Omega }}$$

(218)

where ${\tilde{M}}_T^0$ is defined in (207).

Using (23), (28) and (218) we derive successfully from (217) that

$$\tilde{\kappa }\parallel {\theta }_i{\parallel }_{1,\mathsf{\Omega }}\le ({\beta }_1/{\alpha }_2\lambda ){\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}+\frac{{\mu }_0{\tilde{M}}_T^0}{{\alpha }_2\lambda },$$

(219)

$$|\tilde{\kappa }({\xi }_i·\nabla T_i,{\theta }_i)|\le {\gamma }_2\tilde{\kappa }{M}_T^0\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}{\parallel {\theta }_i\parallel }_{1,\mathsf{\Omega }}\le$$

$$\le \frac{{\beta }_1{\gamma }_2}{{\alpha }_2\lambda }{M}_T^0\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}^2+{\mu }_0{\gamma }_2{M}_T^0{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}\frac{{\tilde{M}}_T^0}{{\alpha }_2\lambda }.$$

(220)

In virtue of (185)–(187), (219), (220) and (145), we infer from (216) that

$$({\delta }_0\nu /2)\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}^2+({\delta }_1{\nu }_m/2)\kappa {\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$\le [{\delta }_0\nu -{\gamma }_0{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0+\frac{{\beta }_1{\gamma }_2}{{\alpha }_2\lambda }{M}_T^0]{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}^2+$$

$$+[{\delta }_1{\nu }_m-{\gamma }_1{M}_{\mathbf{u}}^0-{\gamma }_1(\sqrt{\kappa }/2)M_{\mathbf{H}}^0]\kappa {\parallel {\eta }_i\parallel }_{1,\mathsf{\Omega }}^2\le$$

$$\le {\mu }_0{\gamma }_2{M}_T^0{\parallel {\xi }_i\parallel }_{1,\mathsf{\Omega }}\frac{{\tilde{M}}_T^0}{{\alpha }_2\lambda }.$$

(221)

From (221), arguing as in derivation of the estimates (192), (193), we derive the following estimates:

$$\parallel {\xi }_i{\parallel }_{1,\mathsf{\Omega }}\le \frac{2}{{\delta }_0\nu }\frac{{\mu }_0{\gamma }_2{M}_T^0{\tilde{M}}_T^0}{{\alpha }_2\lambda }\le \frac{2{\mu }_0{\tilde{M}}_T^0\mathcal{R}a}{{\beta }_1},$$

(222)

$$\parallel {\eta }_i{\parallel }_{1,\mathsf{\Omega }}\le \frac{2}{{\delta }_1{\nu }_m}\frac{{\mu }_0{\gamma }_2{M}_T^0{\tilde{M}}_T^0}{\kappa {\alpha }_2\lambda }\le \frac{2{\mu }_0\sqrt{\mathcal{P}m}{\tilde{M}}_T^0\mathcal{R}a}{{\beta }_1\sqrt{\kappa }},$$

(223)

$$\tilde{\kappa }{\parallel {\theta }_i\parallel }_{1,\mathsf{\Omega }}\le \frac{{\beta }_1}{{\alpha }_2\lambda }2{\mu }_0{\tilde{M}}_T^0\left(\frac{\mathcal{R}a}{{\beta }_1}\right)+\frac{{\mu }_0{\tilde{M}}_T^0}{{\alpha }_2\lambda }\le \frac{2{\mu }_0{\tilde{M}}_T^0(\mathcal{R}a+0.5)}{{\alpha }_2\lambda }.$$

(224)

Besides, from (22), (207) and (223) it follows that

$$|{\nu }_1(\mathbf{j},\mathrm{rot}\eta )|\le 4C_1{\mu }_0{\nu }_1\sqrt{\mathcal{P}m}\mathcal{R}a\frac{{\tilde{M}}_T^0}{{\beta }_1\sqrt{\kappa }}{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}={\mu }_{0}a{\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}.$$

(225)

Using (19)–(25), (159), (160), (164), (210) and (222)–(224), we have

$$|((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)-\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)+$$

$$+\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})|+\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)\le$$

$$\le {\gamma }_0{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2(\parallel {\xi }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\xi }_2{\parallel }_{1,\mathsf{\Omega }})+{\gamma }_1{\kappa \parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2(\parallel {\xi }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\xi }_2{\parallel }_{1,\mathsf{\Omega }})+$$

$$+{\gamma }_1{\kappa \parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}(\parallel {\eta }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\eta }_2{\parallel }_{1,\mathsf{\Omega }})+$$

$$+{\gamma }_2\tilde{\kappa }{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel T\parallel }_{1,\mathsf{\Omega }}(\parallel {\theta }_1{\parallel }_{1,\mathsf{\Omega }}+\parallel {\theta }_2{\parallel }_{1,\mathsf{\Omega }})\le$$

$$\le 20{\mu }_0{\tilde{M}}_T^0\left[{\gamma }_0\frac{\mathcal{R}a}{{\beta }_1}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}^2+{\gamma }_1\kappa \frac{\mathcal{R}a}{{\beta }_1}{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}^2\right.+$$

$$\left.+{\gamma }_1\kappa \frac{\sqrt{\mathcal{P}m}\mathcal{R}a}{{\beta }_1\sqrt{\kappa }}{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel \mathbf{H}\parallel }_{{\mathcal{H}}_m^r\left(\mathsf{\Omega }\right)}+{\gamma }_2\frac{(\mathcal{R}a+0.5)}{{\alpha }_2\lambda }{\parallel \mathbf{u}\parallel }_{1,\mathsf{\Omega }}{\parallel T\parallel }_{1,\mathsf{\Omega }}\right]\le$$

$$\le \frac{20{\mu }_0{\tilde{M}}_T^0}{{\beta }_1}\left[{\gamma }_0{C}_{\mathbf{u}}^2\mathcal{R}a+{\gamma }_1{C}_{\mathbf{H}}^2\mathcal{R}a+{\gamma }_1{C}_{\mathbf{u}}{C}_{\mathbf{H}}\sqrt{\mathcal{P}m}\mathcal{R}a+{\gamma }_2{C}_{\mathbf{u}}{C}_T{\mathcal{R}}_3(\mathcal{R}a+0.5)\right]\times$$

$$\times \left[{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+{\kappa \parallel q\parallel }_{s,\mathsf{\Gamma }}^2+{\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2+{(d_3/d_2)}^2{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+{(d_4/d_2)}^2{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2]\right]\le$$

$$\le (1-\epsilon ){\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+(1-\epsilon ){\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+$$

$$+(1-\epsilon ){\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+(1-\epsilon ){\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2+{\mu }_0{a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{\mu }_{0}b\kappa {\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2.$$

(226)

Taking into account (225), (226) from (215) we arrive at

$${\mu }_0{(\parallel T\parallel }_Q^2-{(T,T_d)}_Q)\le -((\mathbf{u}·\nabla )\mathbf{u},{\xi }_1+{\xi }_2)+\kappa (\mathrm{rot}\mathbf{H}\times \mathbf{H},{\xi }_1+{\xi }_2)-$$

$$-\kappa (\mathrm{rot}({\eta }_1+{\eta }_2)\times \mathbf{H},\mathbf{u})-\tilde{\kappa }(\mathbf{u}·\nabla T,{\theta }_1+{\theta }_2)-{\nu }_1(\mathbf{j},\mathrm{rot}\eta )-$$

$$-{\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-{\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-{\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-{\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\le$$

$$\le -\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2+$$

$$+{\mu }_0{a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{\mu }_{0}b\kappa {\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2.$$

(227)

Omitting the nonpositive term $-\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2-\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2-\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2-\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2$, we derive from (227) that

$${\parallel T\parallel }_Q^2-{\parallel T\parallel }_Q\parallel T_d{\parallel }_Q\le {a\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}+{b\kappa \parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}^2\equiv {\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}).$$

(228)

Solving the quadratic inequality (228) we have

$${\parallel T\parallel }_Q\le T_d{\parallel }_Q+{\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }}).$$

(229)

Since $T=T_1-T_2$, $T_d=T_d^{\left(1\right)}-T_d^{\left(2\right)}$, $\mathbf{j}={\mathbf{j}}_1-{\mathbf{j}}_2$, Lemma 7 is proved. □

If $Q=\mathsf{\Omega }$ and ${\mathbf{j}}_1={\mathbf{j}}_2$, the estimate (211) has the sense of stability estimate in $L^2\left(\mathsf{\Omega }\right)$ norm for the component $\widehat{T}$ of the solution $(\widehat{\mathbf{u}}$, $\widehat{\mathbf{H}}$, $\widehat{p}$, $\widehat{T}$, $\widehat{\mathbf{g}}$, $\widehat{q}$, $\widehat{\psi }$, $\widehat{\chi })$ of problem (206) with respect to small disturbances of function $T_d\in L^2\left(\mathsf{\Omega }\right)$ in the norm of $L^2\left(\mathsf{\Omega }\right)$. Additionally, if $T_d^{\left(1\right)}=T_d^{\left(2\right)}$ and ${\mathbf{j}}_1={\mathbf{j}}_2$ we conclude from (229) that $T_1=T_2$ in Q. This equality and (227) yield that ${\mathbf{g}}_1={\mathbf{g}}_2$, $q_1=q_2$, ${\psi }_1={\psi }_2$, ${\chi }_1={\chi }_2$. From this fact and (159), (160), (164) and (170) follows that ${\mathbf{u}}_1={\mathbf{u}}_2$, ${\mathbf{H}}_1={\mathbf{H}}_2$, $T_1=T_2$ and $p_1=p_2$ in $\mathsf{\Omega }$. The latter is equivalent to the uniqueness of the solution to the problem (206).

It remains to prove the stability estimates for differences $\mathbf{u}={\mathbf{u}}_1-{\mathbf{u}}_2$, $\mathbf{H}={\mathbf{H}}_1-{\mathbf{H}}_2$, $T=T_1-T_2$ and $p=p_1-p_2$, which are analogous to estimates (200)–(205). To this end, we turn to the inequality (227). Using (228), (229) one can easily deduce from (227) that

$$\epsilon {\mu }_1{\parallel \mathbf{g}\parallel }_{1/2,\mathsf{\Gamma }}^2+\epsilon {\mu }_2{\parallel q\parallel }_{s,\mathsf{\Gamma }}^2+\epsilon {\mu }_3{\parallel \psi \parallel }_{1/2,{\mathsf{\Gamma }}_D}^2+\epsilon {\mu }_4{\parallel \chi \parallel }_{{\mathsf{\Gamma }}_N}^2\le$$

$$\le {\mu }_0{(-\parallel T\parallel }_Q^2+{\parallel T\parallel }_Q\parallel T_d{\parallel }_Q)+{\mu }_0{\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})\le {\mu }_0\parallel T_d{\parallel }_Q^2+{\mu }_0{\phi }^2{(\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})\le {\mu }_0{\mathsf{\Delta }}^2,$$

(230)

where $\mathsf{\Delta }$ is defined in (211). From (159), (160), (164), (170) and (230), we come to the required stability estimates which formally coincide with estimates (200)–(205). Let us describe the obtained result as the following theorem where one should set $\mathsf{\Delta }=\parallel T_d^{\left(1\right)}-T_d^{\left(2\right)}{\parallel }_Q+{\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})$.

Theorem 9.

Let parameters a, b, ${\tilde{M}}_T^0$ and function $\phi :[0,\infty )\to [0,\infty )$ be defined by relations (207)–(209) and let assumptions of Lemma 7 be fulfilled. Then, the stability estimates (200)–(205) for problem (206) hold where $\mathsf{\Delta }=\parallel T_d^{\left(1\right)}-T_d^{\left(2\right)}{\parallel }_Q+{\phi (\parallel \mathbf{j}\parallel }_{\mathsf{\Omega }})$.

The uniqueness and stability estimates of the solution of the control problem (75) can be investigated according to a similar scheme in the case when the functional $I\left(\mathbf{x}\right)$ is replaced by one of the other functionals $I_k\left(\mathbf{x}\right)$ in (79) at $k=2,4,\dots ,7$. The authors leave the study of these cases to the reader.

7. Discussion

In this paper we have continued to develop the direction started in our previous works and developed a mathematical apparatus for the study of control problems for a more complex MHD-Boussinesq model, which takes into account the influence of temperature and electromagnetic effects on the motion of electro- and heat-conducting fluid in three-dimensional bounded domains. The development of the mentioned mathematical apparatus was accompanied by certain difficulties caused by the complexity of the considered mathematical MHD-Boussinesq model (1)–(3). To simplify the process of investigation of the control problems under consideration for model (1)–(3), we used the scheme of investigation of control problems for stationary MHD models developed in previous works. This scheme consists of the following six steps (see Alekseev [14]).

(1)

Statement of the main problem (1)–(5).

(2)

Proof of the existence of the weak solution of the problem (1)–(5) and its reduction to the operator formulation.

(3)

Formulation of the general control problem and proof of its solvability.

(4)

The derivation of the necessary conditions of optimality in the form of the Euler–Lagrange equation for the adjoint state and the minimum principle for the controls.

(5)

Establishing additional properties of optimal solutions.

(6)

Derivation of stability estimates of solutions for concrete control problems.

We emphasize that this scheme was developed to study control problems for the classical MHD model, which is described by relations (1), (2) when $b=0$. But this scheme is also applicable for the study of control problems for the MHD-Boussinesq model (1)–(3) considered in this paper, as well as for other hydrodynamic models of continuum mechanics. We stress that the described scheme is also useful for potential readers of this paper.

We note that the main result obtained by us is the stability estimates of optimal solutions of control problems (172) and (206) having the form (200)–(205). They are given in the formulations of Theorems 8 and 9 in Section 6. Analysis of these estimates shows that small perturbations of the input data in the form of the cost functional $I_1$ or $I_3$ and the density of outward currents $\mathbf{j}$ lead to small perturbations of the solutions of the corresponding control problems (172) or (206). Moreover, in the case when perturbations are absent, i.e., ${\tilde{\mathbf{u}}}_d={\mathbf{u}}_d$ or ${\tilde{T}}_d=T_d$ and $\tilde{\mathbf{j}}=\mathbf{j}$, the right-hand sides of estimates (200)–(205), and hence the left-hand sides, vanish. The latter means the uniqueness of the solution of the control problems (172) and (206). Thus, taking into account Theorem 3 and Corollary 1 about the existence of solutions to problems (172) and (206), we conclude, in turn, that the control problems (172) and (206) are correctly posed.

But it should be noted that we proved the stability and correctness of the control problems under study using a number of conditions, in particular, using conditions (112), (177), which have the sense of conditions of smallness of the input data. Therefore, we have proved local uniqueness and stability theorems for the solutions of the control problems (172) and (206), which are specifics of the nonlinear model (1)–(3) under consideration. We emphasize that similar conditions of smallness of the input data were used in Alekseev [58] when proving the correctness of the control problems for the classical MHD model (1), (2) at $b=0$.

8. Conclusions

We have studied a stationary magnetohydrodynamics Boussinesq model describing the motion of a thermally and electrically conducting fluid in a bounded domain under inhomogeneous boundary conditions for velocity, magnetic field and temperature. Optimal control problems for the mentioned model have been investigated for the case of several boundary controls of hydrodynamic, electromagnetic and temperature types. These problems have been formulated as minimization problems with tracking-type cost functionals. We developed a mathematical apparatus for studying the mentioned control problems. Based on this apparatus we have proved the global solvability of the control problems under consideration, derived optimality systems describing the necessary optimality conditions of the first order and established important additional properties of optimal solutions. Using these properties of optimal solutions, sufficient conditions for the data that ensure the uniqueness and stability of solutions to control problems for concrete cost functionals have been derived. Besides we have proved the stability of these solutions with respect to small perturbations of both the cost functional under minimization and one of the given functions. The obtained stability estimates were written in an explicit and easily interpretable form so that the presented results have a clear mathematical structure and a visual physical sense.