Periodic solution for the magnetohydrodynamic equations with inhomogeneous boundary condition

We show, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of periodic strong solutions for the magnetohydrodynamics's type equations with inhomogeneous boundary conditions. In particular, when the magnetic field h(x,t) is zero, we obtain existence and uniqueness of strong solutions to the Navier-Stokes equations with inhomogeneous boundary conditions.


1
In presence of a free motion of heavy ions (see Schluter [22], [23] and Pikelner [19]), the MHD equation may be reduced to Here, u and h are unknown velocity and magnetic field, respectively; p * is an unknown hydrostatic pressure; w is an unknown function related to the heavy ions (in such way that the density of electric current, j 0 , generated by this motion satisfies the relation rotj 0 = −σ∇w); ρ is the density of mass of the fluid (assumed to be a positive constant); µ > 0 is a constant magnetic permeability of the medium; σ > 0 is a constant electric conductivity; η > 0 is a constant viscosity of the fluid; f is a given external force field.
The initial value problem associated to the system (1) has been studied by several authors.
Lassner [14], by using the semigroup results of Kato and Fujita [9], proved the existence and uniqueness of strong solutions. Boldrini and Rojas-Medar [5], [21] improved this result to global strong solutions by using the spectral Galerkin method. Damázio and Rojas-Medar [8] studied the regularity of weak solutions, Notte-Cuello and Rojas-Medar [17] used an iterative approach to show the existence and uniqueness of the strong solutions. The initial value problem in time dependent domains was studied by Rojas-Medar and Beltrán-Barrios [20] and by Berselli and Ferreira [4].
The periodic problem for the classical Navier-Stokes equations was studied by Serrin [24] using the perturbation method and subsequently by Kato [12] using the spectral Galerkin method.
Following the methodology used by Kato, Notte-Cuello and Rojas-Medar [18] studied the existence and uniqueness of periodic strong solutions with homogeneous boundary conditions for the MDH type equations. In this work it is considered the periodic problem for the MHD equations with inhomogeneous boundary conditions. We prove the existence and the uniqueness of the strong solutions to this system of equations, following the methodology used by Morimoto [16], who presented results of existence and uniqueness of weak solutions to the Navier-Stokes equations and to the Boussinesq equations.

Preliminaries and Results
We begin by recalling certain definitions and facts to be used later in this paper.
The L 2 (Ω)-product and norm are denoted by (, ) and | |, respectively; the L p (Ω)-norm by | | L p , 1 ≤ p ≤ ∞; the H m (Ω)-norm is denoted by H m and the W k,p (Ω)-norm by | | W k,p . Here H m (Ω) = W m,2 (Ω) and W k,p (Ω) are usual Sobolev spaces, If B is a Banach space, we denote L q (0, T ; B) the Banach space of the B-valued functions defined in the interval (0, T) that are L q -integrable in the sense of Bochner.
Let P be the orthogonal projection from (L 2 (Ω)) n onto H obtained by the usual Helmholtz decomposition. Then, the operator A : H → H given by A = −P ∆ with domain D(A) = (H 2 (Ω)) n ∩ V is called the Stokes operator.
In order to obtain regularity properties of the Stokes operator we will assume that Ω is of class C 1,1 [3]. This assumption implies, in particular, that when Au ∈ L 2 (Ω), then u ∈ H 2 (Ω) and u H 2 and |Au| are equivalent norms. Now, let us introduce some functions spaces consisting of τ -periodic functions. For k ≥ 0, k ∈ N, we denote by Then, let us define the norm We denote for 1 ≤ p ≤ ∞, the spaces Similarly, we denote by In particular, H k (τ ; B) = W k,2 (τ ; B), when B is a Hilbert space.
The problem we consider is as follows: Let the given external force f be periodic in t with some periodic τ. Then we try to prove the existence and uniqueness of periodic strong solutions (u, h) of the magnetohydrodynamic equations (1)-(2) with some periodic τ : Now, according to the Gauss theorem, the boundary value β i i = 1, 2, should satisfy the so-called general outflow condition (GOC) If N > 1, the stringent outflow condition (S.O.C), is stronger than G.O.C.
Also, we consider the Sobolev inequality [10], and the inequality due to Giga and Miyakawa [10] Here, we note that if r = n in (5) it follows Lemma 3 If u ∈ D(A θ ) and 0 ≤ θ < β, then are the eigenvalues of the Stokes operator.
Lemma 4 (Simon [25])Let X, B and Y Banach spaces such that X ֒→ B ֒→ Y , where the first embedding is compact and the second is continuous. Then, if T > 0 is finite, we have that the following embedding is compact Our results are the following.
Theorem 6 (Uniqueness) The solution for (1)-(3) given in the above theorem is unique.
The idea of the proof is to use the spectral Galerkin method together with compactness arguments. The principal problem is to obtain the uniform boundedness of certain norms of u k (t) and h k (t) at some point t * . This difficulty was early treated by Heywood [11] to prove the regularity of the classical solutions for Navier-Stokes equations.

Approximate Problem and a priori estimates
We have ( u + B 1 , h + B 2 ) satisfying the following equation: By putting u = u and h = h and rearranging terms, we obtain By using the operator P, the periodic problem (1)-(3) is formulated as follows We , of u and h, respectively, satisfying the following system of ordinary differential equations, To show that system (9) has an unique τ −periodic solution, we consider the following linearized problem: where . It is well know that the linearized system (10) has an unique τ −periodic solution (u k (t), h k (t)) ∈ (C 1 (τ ; V k )) 2 (see for instance, [2], [6]). Consider the map: Φ : . We shall show that Φ has a fixed point by Leray-Schauder Theorem.
We prove that for every (u k , h k ) and λ where C is a positive constant independent of λ.
For λ = 0, (u k , h k ) = (0, 0). Let λ > 0 and assume that λΦ(u k , h k ) = (u k , h k ). Then, from (10), we obtain Summing the above equalities, we obtain We observe that, since λ ≤ 1, we obtain Also, by using the Hölder inequality, we have Now, we use the Lemma 1, to obtain Using the Young inequality, summing the estimates (14), (15) and (16) together with the equality Integrating in t and using the periodicity of (u k , h k ) we have L 4 belong to L 1 (0, τ ) and are independent of k. Whence by the Mean Value Theorem for integrals, there exists t * ∈ [0, τ ] such that On the other hand, by using the Lemma 3, with θ = 0 and β = 1/2, analogously Finally, by integrating again (17) from t * to t + τ, with t ∈ [0, τ ], we obtain (11). As the map Φ is continuous and compact in C 0 (τ ; V k ) we conclude the existence of a fixed point (u k , h k ) for Φ. Observe that (11) holds for this (u k , h k ).
Lemma 7 Let (u k (t), h k (t)) be the solution of (9). Suppose that Proof : Taking A 2γ u k and A 2γ h k as test functions in (9), we get Now, we estimate the right hand side of the above equalities as follows: here we use the Hölder's inequality where we use the Giga-Miyakawa estimate with θ = γ and ρ = (1 + 2γ)/2. Now, we must estimate the L 1 and L 2 terms, where now, we note that Now, we bound the terms of L 2 (A 2γ h k ), here we use θ = 2γ+1 2 and ρ = 3γ 2 in Giga-Miyakawa estimate, Now, summing the above estimates, we get where we put Now, by using the Lemma 3, with θ = 0 and β = 1/2 we have |u k (t * )| ≤ µ −1/2 |∇u k (t * )| and consequently and from (19) and (20), we have Thus, if suppose that M < 1, we obtain at t = t * , Then, we can write . We will prove by contradiction that T * = ∞. In fact, it T * is finite it should follow that ∀t ∈ [t * , T * ).
From above inequations, we can obtain and Therefore, for such a value t = T * , from the estimates of the right hand side of (21) and from (23) we obtain where we use the inequality |A γ u k | ≤ µ −1/2 |A (1+2γ)/2 u k |. Similarly, and Consequently, the above estimate and (21) imply where Thus, in a neighborhood of t = T * it follows which implies T * = ∞. Then, we have since u k (t) and h k (t) are periodical.
To show the convergence of the approximate solutions we shall derive estimates of derivatives of higher order. By Lemma 7, if M is sufficiently small the approximate solutions satisfy with γ = N 4 − 1 2 , where C(M ) denotes a constant depending on M and on norm involving the border function β i (x, t) and independent of k.
Thus, from the above inequality we can write where Then, integrating (26) and recalling the periodicity of ∇u k (t) and ∇h k (t), we have where we have used the Young's inequality, then Finally, applying the Mean Value Theorem for integrals, we have that there exists t * ∈ [0, τ ] such that We will show that Taking φ = u t and φ = h t in Lemma 4, with X = V , Y = B = H, we obtain the desired convergences.
Once these latter convergences are established, it is a standard procedure to take the limit along the previous subsequences in (9), and we conclude that (u, h) is a periodic strong solution of (1)-(3).
Note that the NS equations ∂u ∂t − η ρ ∆u + u · ∇u = f − 1 ρ ∇p * u = β(x, t) on ∂Ω div u = 0 are a particular case of the MHD equations when the magnetic field h is identically zero, in this case when h = 0, we prove existence and uniqueness of periodic strong solutions to the NS equations with inhomogeneous boundary conditions. Also, Morimoto in [16] shows existence and uniqueness of weak solutions with inhomogeneous boundary conditions for the NS equations.