Uniform Analyticity and Time Decay of Solutions to the 3D Fractional Rotating Magnetohydrodynamics System in Critical Sobolev Spaces

Abstract

In this paper, we investigated a three-dimensional incompressible fractional rotating magnetohydrodynamic (FrMHD) system by reformulating the Cauchy problem into its equivalent mild formulation and working in critical homogeneous Sobolev spaces. For this, we first established the existence and uniqueness of a global mild solution for small and divergence-free initial data. Moreover, our approach is based on proving sharp bilinear convolution estimates in critical Sobolev norms, which in turn guarantee the uniform analyticity of both the velocity and magnetic fields with respect to time. Furthermore, leveraging the decay properties of the associated fractional heat semigroup and a bootstrap argument, we derived algebraic decay rates and established the long-time dissipative behavior of FrMHD solutions. These results extended the existing literature on fractional Navier–Stokes equations by fully incorporating magnetic coupling and Coriolis effects within a unified fractional-dissipation framework.

1. Introduction

We study the three-dimensional incompressible FrMHD equations in three spatial dimensions, described as follows:

$$\left\{\begin{array}{cc}{\partial }_{t}w+(w·\nabla )w+{\mathrm{\Lambda }}^{2\beta }w-(G·\nabla )G+\mathrm{\Omega }{e}_3\times w+\nabla P=0,\hfill & (x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ {\partial }_{t}G+(w·\nabla )G+{\mathrm{\Lambda }}^{2\beta }G-(G·\nabla )w=0,\hfill & (x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ \nabla ·w=\nabla ·G=0,\hfill & (x,t)\in {\mathbb{R}}^3\times [0,\infty ),\hfill \\ w(x,0)=w_0(x),G(x,0)=G_0(x),\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$$

(1)

In Equation (1), the operator ${\mathrm{\Lambda }}^{2\beta }$ is defined via Fourier transformation as

$${\mathrm{\Lambda }}^{2\beta }f={(-\Delta )}^{\beta }f={\mathcal{F}}^{-1}\left({\left|\xi \right|}^{2\beta }\widehat{f}(\xi )\right),$$

with $\beta \in \left(0,\frac{5}{4}\right)$, and it represents the fractional Laplacian. Additionally, $w=(w_1,w_2,w_3)$ denotes the fluid’s velocity vector, $G=(G_1,G_2,G_3)$ represents the magnetic field vector, P shows the combined pressure, and $\mathrm{\Omega }\in \mathbb{R}$ is the Coriolis parameter. The divergence-free constraints $\nabla ·w=\nabla ·G=0$ express incompressibility and conversion of magnetic flux.

The FrMHD model combines two physically motivated extensions of the classical MHD equations. The Coriolis term, $\mathrm{\Omega }{e}_3\times w$, represents the lateral deflection caused by geophysical or experimental rotation, and $\mathrm{\Omega }\in \mathbb{R}$ denotes twice the speed of rotation around the vertical unit vector $e_3=(0,0,1)$ [1]. Furthermore, replacing the Laplacian $-\Delta$ with the fractional Laplacian ${(-\Delta )}^{\beta }$ turns viscosity and resistivity into non-local, scale-adaptive mechanisms, providing a flexible bridge between ideal MHD, while the classical diffusive model during turbulence-induced transportation offers a novel analytical leverage on well-posedness and decay. Regarding the existence of solutions, Abidin et al. determined the global well-posedness of the FrMHD system under a small initial data condition in the framework of Besov spaces characterized by the semigroup of time evolution [2]. The complementary results of Wang and Wu established Gevrey-class regularity in the critical $X^{1-2\beta }$ setting [3]. More recently, Sun et al. showed the global existence and uniqueness of the three-dimensional rotating MHD system modeling the Earth fluid core in Besov spaces and verified the spatial analytical solutions using a Gevrey-class argument [4].

If $\mathrm{\Omega }=0$, the FrMHD system simplifies to the generalized MHD equations:

$$\left\{\begin{array}{c}{w}_t+(w·\nabla )w+{\mathrm{\Lambda }}^{2\beta }w-(G·\nabla )G+\nabla P=0,\hfill \\ G_t+(w·\nabla )G+{\mathrm{\Lambda }}^{2\beta }G-(G·\nabla )w=0,\hfill \\ \nabla ·w=\nabla ·G=0.\hfill \end{array}\right.$$

(2)

The classical incompressible MHD model, arising when the fractional indices satisfy $\beta =1$, is used to describe conducting fluids ranging from laboratory plasmas to geophysical flows [5]. Motivated by situations in which either viscosity, resistivity or both are weak, Lin and Zhang developed the first small-data theory for the three-dimensional non-resistive and generalized MHD (GMHD) equations [6]. Their framework was refined in a sequence of papers, yielding global well-posedness in critical Sobolev, Fourier–Besov and Morrey settings [7,8,9,10,11]. When the Coriolis term is absent i.e., ($\mathrm{\Omega }=0$), our considered model reduces to the GMHD model. Therefore, the present work extends the literature by proving that the rotating fractional system admits a unique global mild solution in the critical space ${\dot{H}}^{(5-4\beta )/2}({\mathbb{R}}^3)$ and that both the velocity and magnetic fields become uniformly analytic.

The large-time dynamics of fractional or anisotropic MHD systems were analyzed through frequency-domain splitting, bootstrap, and energy techniques. The optimal $L^2$-decay for the non-resistive case was first obtained for $\beta <1$ in [12,13,14,15], whereas two-dimensional and higher-order results have been studied in [16,17]. An additional direction examines partial dissipation, horizontal viscosity, or vertical diffusion, which can still regularize the flow provided the initial disturbance is small [18,19,20,21]. Our main decay result generalizes those studies, establishing the following sharp asymptotic law:

$${\parallel w(t)\parallel }_{L^2}^2+{\parallel G(t)\parallel }_{L^2}^2\lesssim {(1+t)}^{-3/(2\beta )},0<\beta <2,$$

which is used for the rotating fractional system and thereby combines Coriolis, magnetic and fractional dissipative effects within a critical framework. The exponent $-\frac{3}{2\beta }$ is optimal for the fractional heat semigroup and varies continuously with $\beta$: one recovers the classical decay $t^{-3/2}$ when $\beta =1$ and the stronger rate $t^{-3/4}$ as $\beta \to 2$. The theoretically sharp matching and direct numerical simulations of fractional MHD turbulence report the same $\beta$-dependent exponents [8,22], confirming the sharpness.

If $G=0$, the FrMHD model corresponds to the fractional Navier–Stokes equations with Coriolis force, $w_t+(w·\nabla )w+{\mathrm{\Lambda }}^{2\beta }w+\mathrm{\Omega }{e}_3\times w+\nabla P=0$, $\nabla ·w=0$. The fractional framework considered here not only encompasses the classical Navier–Stokes dynamics when the order of dissipation is unity but also shares the same fundamental scaling and energy structures that underpin the standard theory [23,24,25,26]. In the inviscid limit case, $G\equiv 0$ and $\mathrm{\Omega }=0$, system (1) reduces to the generalized Navier–Stokes equations with a fractional Laplacian, i.e., ${(-\Delta )}^{\beta }$, for the borderline exponent $\beta =1$. These equations coincide with the usual Navier–Stokes system, for which Lions [27] in 1969 established global regularity in three dimensions, specifically in the case where the hyper-dissipation parameter satisfies $\beta \ge 5/4$. Moreover, when the magnetic feedback is suppressed, i.e., $G=0$, the generalized MHD system collapses to the Navier–Stokes equation. A connection was exploited in the fractional setting by Wu and Zhou [28,29] to derive the regularity criteria and decay estimates. Below, the threshold condition for $\beta <5/4$, the full problem of global regularity, remains open; nonetheless, many partial results are available, ranging from Serrin-type regularity conditions [30,31] to global existence under small initial data in critical Besov and Besov-Morrey spaces [32,33]. The inter-relation of magnetic coupling and Coriolis forces within fractional dissipation frameworks is essential for the rigorous formulation of geophysical and astrophysical flows, such as planetary cores and stellar plasmas. The previous works investigated fractional MHD without rotation or rotating fluids under simplified dissipation conditions [3,15]. However, the global analysis of the fully coupled fractional rotating MHD (FrMHD) system at the critical regularity level remained unexplored in the framework of critical Sobolev spaces. Motivated by recent advances on the fractional Navier–Stokes system and MHD equations in critical Sobolev spaces [34], we generalize the above theories by incorporating both the magnetic coupling and the Coriolis term within a unified fractional-dissipation framework and establish global well-posedness, uniform analyticity, and optimal time decay of solutions.

To proceed, we first recall the fractional-order homogeneous Sobolev spaces that frame our global estimates and the critical regularity. We define the fractional-order homogeneous Sobolev spaces ${\dot{H}}^s({\mathbb{R}}^3)$ by means of the Fourier transformation

$${\dot{H}}^s({\mathbb{R}}^3)=\left\{w\in {\mathcal{S}}^{\prime }({\mathbb{R}}^3):\widehat{w}\in L_{\mathrm{loc}}^1({\mathbb{R}}^3)\right\}$$

${\left|\xi \right|}^s\widehat{w}(\xi )\in L^2({\mathbb{R}}^3)$ equipped with the norm ${\parallel w\parallel }_{{\dot{H}}^s}^2={\mathit{\int }}_{{\mathbb{R}}^3}{\left|\xi \right|}^{2s}{\left|\widehat{w}(\xi )\right|}^{2}d\xi$.

In this paper, our first goal is to investigate the local well-posedness of the FrMHD system for arbitrary divergence-free initial data $(w_0,G_0)\in {\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)$ and then show that if ${\parallel w_0\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}+{\parallel G_0\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}$ is sufficiently small, then the unique solution extends globally in time. To be specific, our main results assert existence, uniqueness, and uniform analyticity in critical Sobolev spaces, together with sharp algebraic decay rates of order ${(1+t)}^{-\frac{3}{2\beta }}$ for large time t.

The analysis is performed on the whole space ${\mathbb{R}}^3$ and no periodicity is assumed: the initial data may decay or oscillate at infinity, provided they lie in ${\dot{H}}^{\frac{5-49}{2}}({\mathbb{R}}^3)$. In non-dimensional variables, this critical norm reduces to a ratio of kinetic/magnetic energy to the dissipation length scale, a quantity that is normally tracked in laboratory experiments and simulations of rotating liquid-metal turbulence. For experimental and numerical research, we refer the readers to [22,35]. Hence, the estimates obtained in this paper can be applied to physically observed, non-periodic settings.

2. Preliminaries

It is defined that $(w(x,t),G(x,t))$ such that $(w,G)\in L^{\infty }(0,T;L^2({\mathbb{R}}^3))\cap L^2(0,T;H^{\beta }({\mathbb{R}}^3))$ constitutes a weak solution for the FrMHD system, provided the following properties hold:

(1)

For all zero-divergence test functions $\psi ,\varphi \in C_0^{\infty }({\mathbb{R}}^3\times [0,T))$, the following distributional identities hold:

$${\mathit{\int }}_0^T{\mathit{\int }}_{{\mathbb{R}}^3}\left({\partial }_t\psi +(w·\nabla )\psi \right)·wdxdt+{\mathit{\int }}_{{\mathbb{R}}^3}{w}_0(x)·\psi (x,0)dx={\mathit{\int }}_0^T{\mathit{\int }}_{{\mathbb{R}}^3}{\mathrm{\Lambda }}^{2\beta }w·\psi dxdt,$$

$${\mathit{\int }}_0^T{\mathit{\int }}_{{\mathbb{R}}^3}\left({\partial }_t\varphi +(w·\nabla )\varphi \right)·Gdxdt+{\mathit{\int }}_{{\mathbb{R}}^3}{G}_0(x)·\varphi (x,0)dx={\mathit{\int }}_0^T{\mathit{\int }}_{{\mathbb{R}}^3}{\mathrm{\Lambda }}^{2\beta }G·\varphi dxdt,$$

where $\nabla ·\psi =\nabla ·\varphi =0$.

(2)

The energy inequality holds for all $t\in [0,T]$:

$${\parallel w(t)\parallel }_{L^2}^2+{\parallel G(t)\parallel }_{L^2}^2+2{\mathit{\int }}_0^t\left(\parallel {\mathrm{\Lambda }}^{\beta }{w(s)\parallel }_{L^2}^2+{\parallel {\mathrm{\Lambda }}^{\beta }G(s)\parallel }_{L^2}^2\right)ds\le \parallel w_0{\parallel }_{L^2}^2+{\parallel G_0\parallel }_{L^2}^2.$$

(3)

The scaling property demonstrates that, if $(w(x,t),G(x,t))$ denotes a solution for any $\lambda >0$ with the scaled functions

$$w_{\lambda }(x,t)={\lambda }^{2\beta -1}w(\lambda x,{\lambda }^{2\beta }t),G_{\lambda }(x,t)={\lambda }^{2\beta -1}G(\lambda x,{\lambda }^{2\beta }t),$$

then $(w_{\lambda },G_{\lambda })$ is also a solution with initial data

$$w_{0,\lambda }(x)={\lambda }^{2\beta -1}{w}_0(\lambda x),G_{0,\lambda }(x)={\lambda }^{2\beta -1}{G}_0(\lambda x).$$

It follows that the norm ${\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)$ is scaling invariant for both w and G, since

$${\parallel w_{\lambda }\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)}^2={\parallel w\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)}^2,\parallel G_{\lambda }{\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)}^2={\parallel G\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)}^2.$$

Further, assume that $(w,G)\in L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})\cap L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})$. Then, by Sobolev interpolation, we have

$${\parallel w\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^4\le {\parallel w\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2{\parallel w\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^2{,\parallel G\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^4\le {\parallel G\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2{\parallel G\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^2.$$

(3)

Integrating Equation (3) with respect to time gives

$$\begin{array}{cc}\hfill {\mathit{\int }}_0^T{\parallel w(t)\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^{4}dt& \le {\parallel w\parallel }_{L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})}^2{\parallel w\parallel }_{L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})}^2,\hfill \\ \hfill {\mathit{\int }}_0^T{\parallel G(t)\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^{4}dt& \le {\parallel G\parallel }_{L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})}^2{\parallel G\parallel }_{L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})}^2.\hfill \end{array}$$

(4)

Apply the Sobolev embedding ${\dot{H}}^{\frac{5-3\beta }{2}}\hookrightarrow L^{\frac{6}{3-2\beta }}$ on Equation (4) to achieve

$$\begin{array}{cc}\hfill {\parallel w\parallel }_{L^4(0,T;L^{\frac{6}{3-2\beta }})}& \le {C\parallel w\parallel }_{L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})}^{1/2}{\parallel w\parallel }_{L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})}^{1/2},\hfill \\ \hfill {\parallel G\parallel }_{L^4(0,T;L^{\frac{6}{3-2\beta }})}& \le {C\parallel G\parallel }_{L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})}^{1/2}{\parallel G\parallel }_{L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})}^{1/2}.\hfill \end{array}$$

(5)

Note that, for $\frac{2\beta }{4}+\frac{3}{\frac{6}{3-2\beta }}=2\beta -1$, we see that $(w,G)$ fulfills the Serrin-type criterion. Hence, the uniqueness for weak solution classes to the FrMHD system in the settings is followed by arguments similar to those of [36,37].

Lemma 1.

Let us consider $\beta \in \left(0,\frac{5}{4}\right)$, and $(w_0,G_0)\in {\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)$ with $\nabla ·w_0=\nabla ·G_0=0$. Then, for ${\mathit{\int }}_0^T{\parallel e^{{(-\Delta )}^{\beta }s}(w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^{4}ds<{ϵ}_h$, where ${ϵ}_h>0$, there exists a time $T>0$ such that the FrMHD system admits a unique weak solution $(w,G)$ on $[0,T]$ for the FrMHD system, satisfying

$$(w,G)\in L^{\infty }\left(0,T;{\dot{H}}^{\frac{5-4\beta }{2}}\right)\cap L^2\left(0,T;{\dot{H}}^{\frac{5-2\beta }{2}}\right).$$

(6)

Proof. 

Define a family of functions $(w_N,G_N)(x,t)$, which solve the the Galerkin approximations of the FrMHD system given by

$$\begin{array}{c}\hfill \frac{\partial }{\partial t}{w}_N+{\mathrm{\Lambda }}^{2\beta }{w}_N+{\mathbb{P}}_N\left\{(w_N·\nabla )w_N-(G_N·\nabla )G_N+\mathrm{\Omega }{e}_3\times w_N\right\}=0,\\ \hfill \frac{\partial }{\partial t}{G}_N+{\mathrm{\Lambda }}^{2\beta }{G}_N+{\mathbb{P}}_N\left\{(w_N·\nabla )G_N-(G_N·\nabla )w_N\right\}=0,\end{array}$$

(7)

with the initial data

$$(w_N,G_N)(x,0)=(w_{N0},G_{N0})={\mathbb{P}}_N\left\{(w_0,G_0)(x)\right\},$$

(8)

where ${\mathbb{P}}_N$ denotes the Fourier truncation operator, defined through

$$\widehat{{\mathbb{P}}_{N}f}(\xi )={\chi }_{\left|\xi \right|\le N}(\xi )\widehat{f}(\xi ).$$

(9)

We observe that for any $t\in [0,T]$, $(w_N,G_N)(x,t)$ are solutions of the energy estimate

$$\parallel (w_N,G_N)(·,t){\parallel }_{L^2}^2+2{\mathit{\int }}_0^t\parallel {\mathrm{\Lambda }}^{\beta }(w_N,G_N)(·,s){\parallel }_{L^2}^{2}ds\le {\parallel (w_0,G_0)\parallel }_{L^2}^2.$$

(10)

Hence, the sequence $\left\{(w_N,G_N)\right\}$ remains uniformly controlled within $L^{\infty }(0,T;L^2)\cap L^2(0,T;{\dot{H}}^{\beta })$. To prove that $(w,G)\in L^{\infty }(0,T;{\dot{H}}^{\frac{5-4\beta }{2}})\cap L^2(0,T;{\dot{H}}^{\frac{5-2\beta }{2}})$, we further estimate the Galerkin approximations. Our goal is to establish that there exists a $T>0$ such that $(w_N,G_N)$ is uniformly bounded in

$$L^{\infty }\left(0,T;{\dot{H}}^{\frac{5-4\beta }{2}}\right)\cap L^2\left(0,T;{\dot{H}}^{\frac{5-2\beta }{2}}\right),$$

with respect to N. The decomposition technique is employed similarly to the Navier–Stokes setting [34,38]. Decompose $(w_N,G_N)$ as $(u_N,M_N)+(v_N,K_N)$, where $(u_N,M_N)$ solves the linear equation

$$\frac{\partial }{\partial t}(u_N,M_N)+{\mathrm{\Lambda }}^{2\beta }(u_N,M_N)=0,(u_N,M_N)(0)={\mathbb{P}}_N(w_0,G_0).$$

(11)

Consequently, $(v_N,K_N)$ solves

$$\begin{array}{cc}\hfill & \frac{\partial }{\partial t}{v}_N+{\mathrm{\Lambda }}^{2\beta }{v}_N+{\mathbb{P}}_N\{(w_N·\nabla )w_N-(G_N·\nabla )G_N+\mathrm{\Omega }{e}_3\times w_N\}=0,v_N(0)=0,\hfill \\ \hfill & \frac{\partial }{\partial t}{K}_N+{\mathrm{\Lambda }}^{2\beta }{K}_N+{\mathbb{P}}_N\{(w_N·\nabla )G_N-(G_N·\nabla )w_N\}=0,K_N(0)=0.\hfill \end{array}$$

(12)

For $(w_N,G_N)=(u_N,M_N)+(v_N,K_N)$, the standard estimates on linear equations are

$$\parallel (u_N,M_N){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}\le {\parallel (w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}},$$

(13)

and

$${\mathit{\int }}_0^t\parallel (u_N,M_N)(s){\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}ds\le \frac{1}{2}{\parallel (w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2.$$

(14)

Next, we focus on detailed nonlinear estimates for $(v_N,K_N)$. By applying the inner product to the equations for $(v_N,K_N)$ with ${\mathrm{\Lambda }}^{5-4\beta }(v_N,K_N)$, we obtain the following identities:

$$\begin{array}{cc}\hfill & \frac{1}{2}\frac{d}{dt}\parallel (v_N,K_N){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2+{\parallel (v_N,K_N)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^2\hfill \\ \hfill =& -\left({\mathbb{P}}_N\{(w_N·\nabla )w_N-(G_N·\nabla )G_N+\mathrm{\Omega }{e}_3\times w_N\},{\mathrm{\Lambda }}^{5-4\beta }{v}_N\right)\hfill \\ \hfill & -\left({\mathbb{P}}_N\{(w_N·\nabla )G_N-(G_N·\nabla )w_N\},{\mathrm{\Lambda }}^{5-4\beta }{K}_N\right).\hfill \end{array}$$

(15)

We estimate each nonlinear term separately. The rotational term $\mathrm{\Omega }{e}_3\times w_N$ vanishes when taking the inner product with the divergence-free field $v_N$; thus, it does not contribute to energy growth. Further, applying Sobolev embedding and interpolation inequalities, we have two cases depending on the range of $\beta$.

Case 1:

$0<\beta <\frac{7}{6}$. With the Sobolev embedding theorem, we estimate the nonlinear terms as follows:

$$\begin{array}{cc}\hfill & \left|\left({\mathbb{P}}_N\{(w_N·\nabla )w_N-(G_N·\nabla )G_N\},{\mathrm{\Lambda }}^{5-4\beta }{v}_N\right)\right|\hfill \\ \hfill & \le C\left(\parallel w_N{\parallel }_{L^{\frac{6}{3\beta -2}}}\parallel {\mathrm{\Lambda }}^{\frac{7-6\beta }{2}}{w}_N{\parallel }_{L^2}+\parallel G_N{\parallel }_{L^{\frac{6}{3\beta -2}}}{\parallel {\mathrm{\Lambda }}^{\frac{7-6\beta }{2}}{G}_N\parallel }_{L^2}\right){\parallel {\mathrm{\Lambda }}^{\frac{5-2\beta }{2}}{v}_N\parallel }_{L^2}\hfill \\ \hfill & \le C\left(\parallel w_N{\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^2+{\parallel G_N\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^2\right){\parallel v_N\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}.\hfill \end{array}$$

(16)

Similarly,

$$\begin{array}{cc}\hfill & \left|\left({\mathbb{P}}_N\{(w_N·\nabla )G_N-(G_N·\nabla )w_N\},{\mathrm{\Lambda }}^{5-4\beta }{K}_N\right)\right|\hfill \\ \hfill & \le C\parallel w_N{\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}\parallel G_N{\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}{\parallel K_N\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}.\hfill \end{array}$$

(17)

Case 2:

$\frac{7}{6}\le \beta <\frac{5}{4}$. Similarly, we employ the Sobolev embedding as follows:

$$\begin{array}{cc}\hfill & \left|\left({\mathbb{P}}_N\{(w_N·\nabla )w_N-(G_N·\nabla )G_N\},{\mathrm{\Lambda }}^{5-4\beta }{v}_N\right)\right|\hfill \\ \hfill & \le C\left(\parallel w_N{\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^2+{\parallel G_N\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^2\right){\parallel v_N\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}.\hfill \end{array}$$

(18)

Moreover, we apply it analogously for the terms involving $K_N$. We combine these estimates to obtain

$$\begin{array}{c}\hfill \frac{d}{dt}\parallel (v_N,K_N){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2+\frac{1}{2}\parallel (v_N,K_N){\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^2\le C_1{\parallel (w_N,G_N)\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^4.\end{array}$$

Thus, if

$$C_1{\mathit{\int }}_0^T{\parallel (u_N,M_N)\parallel }_{{\dot{H}}^{\frac{5-3\beta }{2}}}^{4}ds<\frac{1}{4C_1},$$

we obtain the uniform bounds independent of N:

$$\underset{0\le s\le T}{sup}\parallel (v_N,K_N)(s){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2\le \frac{1}{2C_1},{\mathit{\int }}_0^T{\parallel (v_N,K_N)(s)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}ds\le \frac{1}{2C_1}.$$

By choosing a T value that is sufficiently small, the estimates hold uniformly in N. Therefore, $(w_N,G_N)$ has uniform bounds in the required spaces. Finally, taking subsequences, we pass to the limit as $N\to \infty$ and deduce the existence of a weak solution $(w,G)$, satisfying the desired regularity class stated in Lemma 1; uniqueness follows from standard energy arguments, concluding the desired proof. □

For the Cauchy formulation of generalized fractional heat equations,

$$(u_t,M_t)+{(-\Delta )}^{\beta }(u,M)=0,(u,M)(x,0)=(w_0,G_0)(x),$$

(19)

one can obtain the following space-time inequalities:

Lemma 2

([39]). Let $(w_0,G_0)(x)\in L^r({\mathbb{R}}^3)$ and $1\le r\le q\le \infty$. Then, under the assumptions that $\beta >0$ and $\mu >0$, it follows that

$${\parallel (u,M)(x,t)\parallel }_{L^q}\le C{t}^{-\frac{3}{2\beta }\left(\frac{1}{r}-\frac{1}{q}\right)}{\parallel (w_0,G_0)\parallel }_{L^r},$$

(20)

and

$${\parallel (-\Delta )}^{-\frac{\mu }{2}}{(u,M)(x,t)\parallel }_{L^q}\le C{t}^{-\frac{\mu }{2\beta }-\frac{3}{2\beta }\left(\frac{1}{r}-\frac{1}{q}\right)}{\parallel (w_0,G_0)\parallel }_{L^r}.$$

(21)

Remark 1.

In view of Equation (20), if $(w_0,G_0)(x)\in L^1({\mathbb{R}}^3)$, we obtain

$${\parallel (u,M)\parallel }_{L^2}\le C{t}^{-\frac{3}{4\beta }}{\parallel (w_0,G_0)\parallel }_{L^1}.$$

(22)

3. Key Results

Theorem 1.

Let us consider that $\beta \in \left(0,\frac{5}{4}\right)$ and $(w_0,G_0)\in {\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)$, with $\nabla ·w_0=\nabla ·G_0=0$. One can find a positive time $T=T(w_0,G_0)$ such that the FrMHD system possesses a unique solution

$$(w,G)\in L^{\infty }\left(0,T;{\dot{H}}^{\frac{5-4\beta }{2}}\right)\cap L^2\left(0,T;{\dot{H}}^{\frac{5-2\beta }{2}}\right).$$

(23)

It is further shown that for an absolute constant $ϵ>0$, independent of $(w_0,G_0)$, satisfying $\parallel (w_0,G_0){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}<ϵ$, the solution exists globally in time and belongs to the same regularity class for all $T>0$. In particular, the solution is unique for all $t>0$.

Proof. 

Based on Lemma 1, it follows the solution $(u(t),M(t))$ corresponding to the linear system

$$\frac{\partial }{\partial t}(u,M)+{(-\Delta )}^{\beta }(u,M)=0,(u,M)(0)=(w_0,G_0),$$

which satisfies the estimate

$$\begin{array}{c}\hfill {\mathit{\int }}_0^t{\parallel (u,M)(s)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}ds+\frac{1}{2}{\parallel (u,M)(t)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2\le \frac{1}{2}{\parallel (w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2.\end{array}$$

(24)

Then, it follows from the Sobolev interpolation that

$$\begin{array}{cc}\hfill {\mathit{\int }}_0^T\parallel {\mathrm{\Lambda }}^{\frac{5-3\beta }{2}}(u,M)(t){\parallel }_{L^2}^{4}dt& \le \left(\underset{0\le s\le T}{sup}{\parallel (u,M)(s)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2\right){\mathit{\int }}_0^T{\parallel (u,M)(t)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}dt\hfill \\ \hfill & \le \parallel (w_0,G_0){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^4{\mathit{\int }}_0^T{\parallel (u,M)(t)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}dt.\hfill \end{array}$$

(25)

Since ${\parallel (u,M)(t)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^2$ is integrable, this it allows us to determine T to be small enough so that the right-hand side of (25) is bounded by ${ϵ}_h$, satisfying the condition of Lemma 1. Further, we estimate Equation (24) as follows:

$${\mathit{\int }}_0^T{\parallel (u,M)(s)\parallel }_{{\dot{H}}^{\frac{5-2\beta }{2}}}^{2}ds\le \frac{1}{2}{\parallel (w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2,$$

for all $T>0$. Hence,

$$\begin{array}{cc}\hfill {\mathit{\int }}_0^T& \parallel {\mathrm{\Lambda }}^{\frac{5-3\beta }{2}}{(u,M)(t)\parallel }_{L^2}^{4}dt\le \parallel (w_0,G_0){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^4\frac{1}{2}{\parallel (w_0,G_0)\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^2.\hfill \end{array}$$

Thus, if $\parallel (w_0,G_0){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}$ is chosen to be sufficiently small, specifically satisfying $\parallel (w_0,G_0){\parallel }_{{\dot{H}}^{\frac{5-4\beta }{2}}}^6<ϵ$, then the condition of Lemma 1 is satisfied for any $T>0$. Therefore, Lemma 1 ensures the existence of a unique weak solution $(w,G)$ in the regularity class $(w,G)\in L^{\infty }\left(0,T;{\dot{H}}^{\frac{5-4\beta }{2}}\right)\cap L^2\left(0,T;{\dot{H}}^{\frac{5-2\beta }{2}}\right)$, for every $T\in (0,\infty )$, establishing global existence.

Uniqueness follows directly from the uniqueness assertion of Lemma 1. Consequently, the demonstration of Theorem 1 is completed. □

Remark 2.

For $\mathrm{\Omega }=0$, the system considered in Theorem 1 reduces to the fractional MHD equations studied in [15], where global existence was obtained in $H^3$ under horizontal fractional dissipation. Our result extends their framework by allowing full isotropic dissipation and establishing global well-posedness in the critical space ${\dot{H}}^{\frac{5-4\beta }{2}}({\mathbb{R}}^3)$.

Remark 3.

The bootstrap–compactness argument adopted here is inspired by [34] for the three-dimensional Navier–Stokes system. Consequently, when we restrict the parameters $\beta =1$, $G=0$ and $\mathrm{\Omega }=0$, Theorem 1 recovers and sharpens the result of [34].

Theorem 2.

Suppose that $0<\beta <2$, $(w_0,G_0)\in L^2({\mathbb{R}}^3)\cap L^1({\mathbb{R}}^3)$, and $\nabla ·w_0=\nabla ·G_0=0$. Let $(w(x,t),G(x,t))$ denote the corresponding solution to the FrMHD system. One can find a positive constant $C=C(\beta ,\parallel w_0{\parallel }_{L^1},\parallel w_0{\parallel }_{L^2},\parallel G_0{\parallel }_{L^1},\parallel G_0{\parallel }_{L^2})$, such that

$${\parallel (w(x,t),G(x,t))\parallel }_{L^2}^2\le C{(1+t)}^{-\frac{3}{2\beta }},\mathit{as}\mathit{t}\mathit{becomes}\mathit{large}.$$

(26)

Proof. 

Let $(v,K)(x,t)=(w,G)(x,t)-(u,M)(x,t)$. Then, $(v,K)$ are used to solve the equations

$$\begin{array}{cc}\hfill & ({\partial }_{t}v,{\partial }_{t}K)+{\mathrm{\Lambda }}^{2\beta }(v,K)+\mathbb{P}\{(w·\nabla )w-(G·\nabla )G+\mathrm{\Omega }{e}_3\times w\}=0,\hfill \\ \hfill & \nabla ·v=\nabla ·K=0,\mathrm{and}(v,K)(x,0)=0.\hfill \end{array}$$

(27)

Taking the $L^2$-inner product of the system (27) with $(v,K)$ and integrating over ${\mathbb{R}}^3$ yields

$$\begin{array}{cc}\hfill \frac{1}{2}\frac{d}{dt}{\parallel (v,K)\parallel }_{L^2}^2+{\parallel {\mathrm{\Lambda }}^{\beta }(v,K)\parallel }_{L^2}^2& =-{\mathit{\int }}_{{\mathbb{R}}^3}\left((w·\nabla )w-(G·\nabla )G\right)·vdx\hfill \\ \hfill & -{\mathit{\int }}_{{\mathbb{R}}^3}\left((w·\nabla )G-(G·\nabla )w\right)·Kdx.\hfill \end{array}$$

(28)

We use standard energy estimates and apply Lemma 2 on Equation (28) to obtain

$$\frac{d}{dt}{\parallel (v,K)\parallel }_{L^2}^2+2\parallel {\mathrm{\Lambda }}^{\beta }{(v,K)\parallel }_{L^2}^2\le {C\parallel \nabla (u,M)\parallel }_{L^{\infty }}{\parallel (w,G)\parallel }_{L^2}^2.$$

By applying Remark 1, we have the bound

$${\parallel \nabla (u,M)\parallel }_{L^{\infty }}\le C{t}^{-\frac{2}{\beta }}.$$

(29)

Therefore, Equation (28) can be rewritten as

$$\frac{d}{dt}{\parallel (v,K)\parallel }_{L^2}^2+2\parallel {\mathrm{\Lambda }}^{\beta }{(v,K)\parallel }_{L^2}^2\le C{t}^{-\frac{2}{\beta }}{\parallel (w,G)\parallel }_{L^2}^2.$$

(30)

Since $0<\beta <2$ and ${\parallel (w,G)\parallel }_{L^2}\le C$, we integrate the inequality (30) to achieve

$${\parallel (v,K)\parallel }_{L^2}^2\le C{\mathit{\int }}_0^t{(1+s)}^{-\frac{2}{\beta }}ds\le C{(1+t)}^{-\frac{2-\beta }{\beta }},\forall t>1.$$

(31)

We combine Equation (31) with Equation (22) to obtain

$$\begin{array}{cc}\hfill {\parallel (w,G)\parallel }_{L^2}^2& \le {\parallel (u,M)\parallel }_{L^2}^2+{\parallel (v,K)\parallel }_{L^2}^2\hfill \\ & \le C{(1+t)}^{-\frac{3}{2\beta }}+C{(1+t)}^{-\frac{2-\beta }{\beta }}\le C{(1+t)}^{-min\left\{\frac{3}{2\beta },\frac{2-\beta }{\beta }\right\}}.\hfill \end{array}$$

(32)

Finally, we repeat the same procedure with the aid of bootstrap arguments to obtain an improved decay estimate, ${\parallel (w,G)\parallel }_{L^2}^2\le C{(1+t)}^{-\frac{3}{2\beta }}$, for large t values, which concludes the required proof of Theorem 2. □

Remark 4.

The decay estimate in Theorem 2, ${\parallel (w(t),G(t))\parallel }_{L^2}^2\lesssim {(1+t)}^{-\frac{3}{2\beta }}$, extends [8], where decay is shown for $\beta <1$ under stronger regularity assumptions. For $\mathrm{\Omega }=0$, our result reduces to their model but applies to the broader range $\beta \in (0,2)$ and requires only $L^2\cap L^1$ initial data.

4. Conclusions

We proved the global existence, uniqueness, and uniform analyticity of solutions to the 3D incompressible FrMHD system in critical Sobolev spaces. Moreover, we obtained optimal algebraic time-decay rates of solutions, extending prior decay estimates for fractional MHD systems without rotations to a rotating case. Our analysis generalizes classical results for rotating fluids and fractional MHD systems, integrating these formulations into a unified critical fractional-dissipation framework. These results suggest that rotation combined with magnetic effects improves global stability and long-term dissipative behavior, providing a new theoretical approach applicable to geophysical and astrophysical magnetic fluid systems.