On regularity criteria for MHD system in anisotropic Lebesgue spaces

Kun Cheng; Yong Zeng; Kun Cheng; Yong Zeng

doi:10.3934/era.2023239

Electronic Research Archive

2023, Volume 31, Issue 8: 4669-4682. doi: 10.3934/era.2023239

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Research article

On regularity criteria for MHD system in anisotropic Lebesgue spaces

Kun Cheng ,
Yong Zeng ^,

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received: 29 March 2023 Revised: 14 June 2023 Accepted: 24 June 2023 Published: 30 June 2023

This paper concerns the regularity criteria of the three-dimensional magnetohydrodynamic (MHD) system in anisotropic Lebesgue spaces. Two regularity results were proved under additional assumptions on the horizontal components of the velocity field ${{\bf{u}}}$ and the magnetic field ${{\bf{B}}}$ , or directions of Elsässer's variables ${{\bf{u}}}\pm{{\bf{B}}}$ .

Keywords:

Citation: Kun Cheng, Yong Zeng. On regularity criteria for MHD system in anisotropic Lebesgue spaces[J]. Electronic Research Archive, 2023, 31(8): 4669-4682. doi: 10.3934/era.2023239

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Abstract

1. Introduction

We consider the following MHD system

$\begin{equation} \left\{\begin{array}{l} {\bf{u}}_{t}-\Delta {\bf{u}}+({\bf{u}} \cdot \nabla) {\bf{u}}-({\bf{B}} \cdot \nabla) {\bf{B}}+\nabla\left(p+\frac{1}{2}|{\bf{B}}|^{2}\right) = 0,\\ {\bf{B}}_{t}-\Delta {\bf{B}}+({\bf{u}} \cdot \nabla) {\bf{B}}-({\bf{B}} \cdot \nabla) {\bf{u}} = 0, \\ {\rm{div}} {\bf{u}} = {\rm{div}} {\bf{B}} = 0, \\ {\bf{u}}(x, 0) = {\bf{u}}_{0}, \quad {\bf{B}}(x, 0) = {\bf{B}}_{0}, \end{array}\right. \end{equation}$

(1.1)

in ${\mathbb{R}}^3$ . Here ${\bf u}$ represents the velocity field, ${\bf B}$ represents the magnetic field and $p$ represents the pressure. ${\bf{u}}_{0}$ and ${\bf{B}}_{0}$ are given initial datum satisfying ${\rm{div}} {\bf{u}}_{0} = {\rm{div}} {\bf{B}}_{0} = 0$ in ${\mathbb{R}}^3$ .

The MHD system has been extensively studied. For studies on the existence of weak and strong solutions, see ^[1,2]. Sermange and Teman ^[2] also studied the smoothness of strong solutions and the so-called squeezing property of the trajectories. The regularity criteria of weak solutions were studied by many authors. For the fundamental Serrin-type regularity criteria, we refer to ^[3,4,5,6] and references therein. For more results on MHD and related systems, see ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20]} and references therein.

This paper concerns the regularity criteria of three-dimensional MHD system in anisotropic Lebesgue space. For the Navier-Stokes equations, Zheng ^[21] first studied anisotropic regularity criterion in terms of one velocity component. Years later, Qian ^[22], Guo et al. ^[23] and Guo et al. ^[24] further studied the regularity condition in anisotropic Lebesgue spaces for the Leary-Hopf weak solutions of Navier-Stokes equations. Guo et al. ^[25] also studied the regularity condition in anisotropic Lebesgue spaces for MHD system. By considering different weights in spatial variables, they proved that if ${\partial}_3 u_3$ and ${{\bf{B}}}$ satisfy certain space-time integrable conditions in anisotropic Lebesgue spaces, then the weak solution is indeed regular.

We are concerned with the regularity criteria of MHD system in anisotropic Lebesgue space under conditions on ${{\bf{u}}}_h = (u_1, u_2, 0)$ and ${{\bf{B}}}_h = (b_1, b_2, 0)$ , or the directions of ${{\bf{u}}}\pm{{\bf{B}}}$ . For the Navier-stokes system, Montgomery-Smith ^[26] proved the logarithmically improved regularity criteria for the Navier-Stokes equations, which says that condition

$\int_0^T\frac{\|{{\bf{u}}}(t)\|_{L^q}^p}{1+\log^+\|{{\bf{u}}}(t)\|_{L^q}}\,dt < \infty {\quad}{\rm{with}}{\quad} \frac{2}{p}+\frac{3}{q} = 1{\quad}{\rm{and}}{\quad} 3 < q < \infty$

implies that ${{\bf{u}}}$ is regular on $(0, T]\times{\mathbb{R}}^3$ . Vasseur ^[27] showed that the condition

${\rm{div}}\left(\frac{\bf u}{|\bf u|}\right) \in L^{p}(0,T;L^{q}({\mathbb{R}}^3))\quad {\rm{with}} \quad \frac{2}{p}+\frac{3}{q}\leq\frac{1}{2}\quad {\rm{and}}\quad 6\leq q \leq \infty$

ensures the smooth of weak solution. In ^[28], Miller extended the Ladyzhenskaya-Prodi-Serrin regularity criterion to the Lebesgue sum space and proved that if the maximum existence time $T$ of the local smooth solution is finite, then, it holds that

$\int_0^T\|{\bf{v}}\|_{L^p}^q\,dt+\int_0^T\|\sigma\|_{L^\infty}\,dt = \infty{\quad}{\rm{with}}{\quad} \frac{2}{q}+\frac{3}{p} = 1{\quad}{\rm{and}}{\quad} 3 < p < \infty,$

where ${{\bf{u}}} = {\bf{v}}+\sigma$ . In ^[29], Wu improves the above results ^[26,27,28] in anisotropic Lebesgue spaces. More precisely, let ${{\bf{u}}}_h = \xi+\sigma$ and ${{\rm{div}}}\left(\frac{{{\bf{u}}}}{|{{\bf{u}}}|}\right) = f+g$ , Wu proved that if $q > 0$ then either

$\int_o^T\frac{\|\xi(t)\|_{L^{\vec{p}}}^q+\|\sigma(t)\|_{L^\infty}^2}{1+\ln(e+\|{{\bf{u}}}_h\|_{L^{\vec{s}}})}\,dt < \infty$

with

$\frac{2}{q}+\sum\limits_{i = 1}^3\frac{1}{p_i} = 1,{\quad} \sum\limits_{i = 1}^3\frac{1}{s_i} = \frac{1}{2},{\quad} 2 < p_i\leq\infty,{\quad} 2 < s_i < \infty, {\quad}{\rm{for }}\;i = 1,2,3$

$\int_0^T\|f\|_{L^{\vec{p}}}^q\,dt+\int_0^T\|g\|_{L^\infty}^4\,dt < \infty$

with

$\frac{2}{q}+\sum\limits_{i = 1}^3\frac{1}{p_i} = \frac{1}{2},{\quad} 2 < p_i\leq\infty$

is sufficient to ensure the smoothness of ${{\bf{u}}}$ .

Motivated by the work of Wu ^[29], we study the regularity of MHD systems in the framework of anisotropic Lebesgue space.

2. Preliminaries and main results

Let us first recall the definition of anisotropic Lebesgue spaces introduced by Benedek and Panzone ^[30].

Definition 2.1. For a given $\vec{p} = \left(p_{1}, p_{2}, p_{3}\right) \in[1, \infty)^{3}$ , the anisotropic Lebesgue space $L^{\vec{p}}\left(\mathbb{R}^{3}\right)$ is defined to be the space consisting of all measurable functions $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ such that the norm

$\|f\|_{L ^{\vec{p}}\left(\mathbb{R}^{3}\right)} = \left(\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}|f(x)|^{p_{1}} \,d x_{1}\right)^{\frac{p_{2}}{p_{1}}} \,d x_{2}\right)^{\frac{p_{3}}{p_{2}}}\, d x_{3}\right)^{\frac{1}{p_{3}}} < \infty.$

We also write the norm as $\|f\|_{L^{p_1}_{1}L^{p_2}_{2}L^{p_3}_{3}}$ .

Now we state our main results as follows.

Theorem 2.2. Suppose that $({\bf u}_{0}, {\bf B}_{0}) \in H^{1}\left(\mathbb{R}^{3}\right)$ and ${{\rm{div}}}{\bf u}_{0} = {{\rm{div}}}{{\bf{B}}}_{0} = 0$ . Let $({\bf u}, {\bf B})$ be a weak solution to (1.1) on $[0, T]$ . Assume that ${\bf u}_{h} = \left(u_{1}, u_{2}, 0\right) = \xi_{1}+\sigma_{1}$ and ${\bf B}_{h} = \left(B_{1}, B_{2}, 0\right) = \xi_{2}+\sigma_{2}$ such that

$\int_{0}^{T} \frac{\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_2\|_{L^{\vec{k}}}^l+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^{2}}{1+\ln (e+\|{\bf u}_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}})} \,d t < \infty,$

with $2 < p_{i}, k_i \leq \infty$ , $\frac{2}{q} = 1-\sum_{i = 1}^{3} \frac{1}{p_{i}} > 0$ , $\frac{2}{l} = 1-\sum_{i = 1}^3\frac{1}{k_i} > 0$ and $2 < s_{i}, r_i < \infty$ , $\sum_{i = 1}^{3} \frac{1}{s_{i}} = \sum_{i = 1}^{3} \frac{1}{r_{i}} = \frac{1}{2}$ , then, $({\bf u}, {\bf B})$ is regular on $(0, T]$ .

Theorem 2.3. Suppose that $({\bf u}_0, {\bf B}_{0})\in H^1({\mathbb{R}}^3)$ and ${{\rm{div}}}{{\bf{u}}}_0 = {{\rm{div}}} {{\bf{B}}}_{0} = 0$ . Let ( ${\bf u}, {\bf B})$ be a weak solution of (1.1) on $[0, T]$ . Assume that ${\rm{div}}(\frac{{\bf u}+{\bf B}}{|{\bf u}+{\bf B}|}) = f_{1}+g_{1}$ and ${\rm{div}}\left(\frac{{\bf u}-{\bf B}}{|{\bf u}-{\bf B}|}\right) = f_{2}+g_{2}$ such that

$\int_0^T(\|f_{1}\|_{L^{\vec{p}}}^q+\|f_2\|_{L^{\vec{r}}}^s)\, d t+\int_0^T(\|g_{1}\|_{L^{\infty}}^4+\|g_2\|_{L^\infty}^4)\, d t < \infty,$

with $2 < p_i, r_i\leq \infty, \frac{2}{q} = \frac{1}{2}-\sum_{i = 1}^3 \frac{1}{p_i} > 0, \frac{2}{s} = \frac{1}{2}-\sum_{i = 1}^3 \frac{1}{r_i} > 0$ , then, $({\bf u}, {\bf B})$ is regular on $(0, T]$ .

Before proving our main results, we recall some fundamental mixed norm inequalities. By successive applications of Hölder's inequality, we immediately have

Lemma 2.4. For $\vec{p} = (p_1, p_2, p_3)$ and $\vec{q} = (q_1, q_2, q_3)$ with

$\frac{1}{p_i}+\frac{1}{q_i} = \frac{1}{2},{\quad} 2\leq p_i,q_i\leq\infty, i = 1,2,3,$

it holds that

$\|fg\|_{L^2({\mathbb{R}}^3)}\leq\|f\|_{L^{\vec{p}}({\mathbb{R}}^3)}\|g\|_{L^{\vec{q}}({\mathbb{R}}^3)}.$

The following proposition is a direct consequence of Lemma 3 in ^[23], see also ^[21,22].

Proposition 2.5. For $p_{1}, p_{2}, p_{3} \in[2, \infty)$ and $0 \leq \sum_{i = 1}^{3} \frac{1}{p_{i}}-\frac{1}{2} \leq 1$ , there exists a positive constant $C$ such that

$\|f\|_{L^{\vec{p}}\left(\mathbb{R}^{3}\right)} \leq C\|\nabla f\|_{L^{2}\left(\mathbb{R}^{3}\right)}^{\frac{3}{2}-\sum\limits_{i}^3 \frac{1}{p_{i}}}\|f\|_{L^{2}\left(\mathbb{R}^{3}\right)}^{\sum\limits_{i}^{3}\frac{1}{p_{i}}-\frac{1}{2}}.$

We also need the following special case of the Sobolev embedding theorem in anisotropic spaces proved in [31, p.181], see also [32, Theorem 2.1].

Lemma 2.6. Let $\vec{s} = \left(s_{1}, s_{2}, \ldots, s_{n}\right) \in[2, \infty]^{n}$ satisfy

$\frac{1}{s_{1}}+\frac{1}{s_{2}}+\cdots+\frac{1}{s_{n}} = \frac{n}{2}-1, \quad s_{n} \in(2, \infty).$

Then, there exists a constant $C = C(\vec{s})$ such that

$\|u\|_{L^{\vec{s}}\left(\mathbb{R}^{n}\right)} \leq C\|u\|_{W^{1,2}\left(\mathbb{R}^{n}\right)}, \quad \forall u \in W^{1,2}\left(\mathbb{R}^{n}\right).$

3. Proof of Theorem 2.2

We first deduce an a priori estimate for $({{\bf{u}}}, {{\bf{B}}})$ .

Multiplying the first equation of (1.1) with ${\bf u}$ and multiplying the second equation of (1.1) with ${\bf B}$ , then integrating by parts over $\mathbb{R}^{3}$ , we get

$\begin{equation} \frac{1}{2} \frac{d}{d t}\|{\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf u}\|_{L^{2}}^{2}-\int ({\bf B}\cdot\nabla ){\bf B} \cdot{\bf u}\,dx = 0 , \end{equation}$

(3.1)

$\begin{equation} \frac{1}{2} \frac{d}{d t}\|{\bf B}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}+\int ({\bf B}\cdot\nabla ){\bf B} \cdot{\bf u}\,dx = 0 . \end{equation}$

(3.2)

Multiplying both sides of the first equation of system (1.1) with $-\Delta {\bf u}$ and the second equation of system (1.1) with $-\Delta {\bf B}$ , then integrating over $\mathbb{R}^{3}$ , we have

$\begin{equation} \begin{gathered} \frac{1}{2} \frac{d}{d t} \int_{\mathbb{R}^{3}}|\nabla {\bf u}|^{2} \,d x+\int_{\mathbb{R}^{3}}|{\nabla}^2 {\bf u}|^{2} \,d x = \int_{\mathbb{R}^{3}} ({\bf u} \cdot \nabla) {\bf u} \cdot\Delta {\bf u} \,d x-\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf B}\cdot\triangle {\bf u} \,d x, \end{gathered} \end{equation}$

(3.3)

$\begin{equation} \begin{gathered} \frac{1}{2} \frac{d}{d t} \int_{\mathbb{R}^{3}}|\nabla {\bf B}|^{2} \,d x+\int_{\mathbb{R}^{3}}|{\nabla}^2 {\bf B}|^{2} \,d x = \int_{\mathbb{R}^{3}} ({\bf u} \cdot \nabla) {\bf B} \cdot\Delta {\bf B}\, d x-\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf u}\cdot\triangle {\bf B} \,d x. \end{gathered} \end{equation}$

(3.4)

Integrating by parts, it follows that

$\begin{align*} &\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf B}\cdot\triangle {\bf u} \,d x = -\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla)( \triangle {\bf u})\cdot{\bf B} \,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}} B_{i}\cdot\partial_{i}{\partial}_{k}{\partial}_k u_{j} \cdot B_{j} \,d x = \sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}} \partial_{i}\partial_{k} u_{j}\cdot( B_{j}\partial_{k} B_{i}+ B_{i}\partial_{k} B_{j})) \,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\cdot\partial_{k}( B_{j}\partial_{k} B_{i}+ B_{i}\partial_{k} B_{j})) \,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\cdot(2\partial_{k} B_{i}\cdot\partial_{k} B_{j}+ B_{j}\partial_{k k} B_{i}+ B_{i}\partial_{k k} B_{j}) \,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}(\partial_{i} u_{j}\cdot B_{j}\cdot\partial_{k k} B_{i}+2\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j})\,d x-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\cdot B_{i}\cdot\partial_{k k} B_{j}\,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}(\partial_{i} u_{j}\cdot B_{j}\cdot\partial_{k k} B_{i}+2\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j})\,d x- \int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf u}\cdot\triangle {\bf B}\,d x. \end{align*}$

Therefore, together with the fact that ${{\rm{div}}}{{\bf{B}}} = 0$ , we have

$\begin{align} &\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf B}\cdot\triangle {\bf u}\, d x+\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf u}\cdot\triangle {\bf B} \,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}(\partial_{i} u_{j}\cdot B_{j}\cdot\partial_{k k} B_{i}+2\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j})\,d x\\ = &\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k}(\partial_{i} u_{j}\cdot B_{j})\cdot\partial_{k} B_{i} \,dx-2\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j}\,d x\\ = &\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k}\partial_{i} u_{j}\cdot B_{j}\cdot\partial_{k} B_{i} \,dx-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j}\,d x \\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k} u_{j}\cdot\partial_{i}( B_{j}\partial_{k} B_{i}) \,dx-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j}\,d x\\ = &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k} u_{j}\cdot\partial_{i} B_{j}\cdot\partial_{k} B_{i} \,dx-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j}\,d x.\\ = &I_1+I_2. \end{align}$

(3.5)

To estimate $I_1$ , we rewrite it as follows

$\begin{align} &-\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k} u_{j}\cdot\partial_{i} B_{j}\partial_{k} B_{i} \,dx\\ = &-\sum\limits_{k = 1}^{3} \sum\limits_{ j = 1}^{3}\sum\limits_{i = 1}^{2}\int_{\mathbb{R}^{3}}\partial_{k} u_{j}\cdot\partial_{i} B_{j}\cdot\partial_{k} B_{i} \,d x-\sum\limits_{k = 1}^{3} \sum\limits_{ j = 1}^{2}\int_{\mathbb{R}^{3}}\partial_{k} u_{j}\cdot\partial_{3} B_{j}\cdot\partial_{k} B_{3} \,d x\\ &+\sum\limits_{k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{k} u_{3}\cdot(\partial_{1} B_{1}+\partial_{2} B_{2})\cdot\partial_{k} B_{3} \,d x\\ \leq& C \int_{\mathbb{R}^{3}}|{\bf B}_{h}|(|\nabla {\bf u}||\nabla^{2} {\bf B}|+|\nabla {\bf B}||\nabla^{2} {\bf u}|)\,d x. \end{align}$

(3.6)

Similarly, we have the following estimate for $I_2$ :

$\begin{equation} -\sum\limits_{i, j, k = 1}^{3} \int_{\mathbb{R}^{3}}\partial_{i} u_{j}\partial_{k} B_{i}\partial_{k} B_{j}\,d x\leq C \int_{\mathbb{R}^{3}}(|{\bf B}_{h}|+|{{\bf{u}}}_h|)(|\nabla {\bf u}||\nabla^{2} {\bf B}|+|\nabla {\bf B}||\nabla^{2} {\bf u}|)\,d x. \end{equation}$

(3.7)

Substituting (3.6) and (3.7) into (3.5), we obtain

$\begin{equation} \begin{aligned} &\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf B}\cdot\triangle {\bf u} \,d x+\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf u}\cdot\triangle {\bf B} \,d x\\ \leq & C\int_{\mathbb{R}^{3}}(|{\bf B}_{h}|+|{{\bf{u}}}_h|)(|\nabla {\bf u}||\nabla^{2} {\bf B}|+|\nabla {\bf B}||\nabla^{2} {\bf u}|)\,d x. \end{aligned} \end{equation}$

(3.8)

Similarly, we can get the following estimates

$\begin{equation} \begin{aligned} \int_{\mathbb{R}^{3}}({\bf u}\cdot\nabla) {\bf u}\cdot\triangle {\bf u} \,d x\leq C \int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf u}||\nabla^{2} {\bf u}|\,d x, \end{aligned} \end{equation}$

(3.9)

and

$\begin{equation} \begin{aligned} \int_{\mathbb{R}^{3}}({\bf u}\cdot\nabla) {\bf B}\cdot\triangle {\bf B} \,d x\leq C \int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf B}||\nabla^{2} {\bf B}|\,d x+C \int_{\mathbb{R}^{3}}|{\bf B}_{h}|(|\nabla {\bf u}||\nabla^{2} {\bf B}|+|\nabla {\bf B}||\nabla^{2} {\bf u}|)\,dx. \end{aligned} \end{equation}$

(3.10)

Combined (3.3)–(3.10), we have

$\begin{align} &\frac{1}{2}\frac{d}{d t}(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})+\|\Delta {\bf u}\|_{L^{2}}^{2}+\|\Delta {\bf B}\|_{L^{2}}^{2}\\ = &\int_{\mathbb{R}^{3}} ({\bf u} \cdot \nabla) {\bf u} \cdot\Delta {\bf u} \,d x+\int_{\mathbb{R}^{3}}({\bf u}\cdot\nabla) {\bf B}\cdot\triangle {\bf B} \,d x\\ &-\int_{\mathbb{R}^{3}} ({\bf B} \cdot \nabla) {\bf B} \cdot\Delta {\bf u} \,d x-\int_{\mathbb{R}^{3}}({\bf B}\cdot\nabla) {\bf u}\cdot\triangle {\bf B} \,d x\\ \leq& C \int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf u}||\nabla^{2} {\bf u}|\,dx +C\int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf B}||\nabla^{2} {\bf B}|\,d x \\ &+C\int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf u}||\nabla^{2} {\bf B}|\,dx+C\int_{\mathbb{R}^{3}}|{\bf u}_{h}||\nabla {\bf B}||\nabla^{2} {\bf u}|\,d x\\ &+C\int_{\mathbb{R}^{3}}|{\bf B}_{h}||\nabla {\bf u}||\nabla^{2} {\bf B}|\,dx+C\int_{\mathbb{R}^{3}}|{\bf B}_{h}||\nabla {\bf B}||\nabla^{2} {\bf u}|\,d x\\ = &J_{1}+J_{2}+J_{3}+J_{4}+J_{5}+J_{6}. \end{align}$

(3.11)

We first estimate $J_{3}$ . By Lemma 2.4, Proposition 2.5 and Young's inequality, we have

$\begin{align*} &J_3 = C\int_{{\mathbb{R}}^3}|{{\bf{u}}}_h||{\nabla}{{\bf{u}}}||{\nabla}^2{{\bf{B}}}|\,dx \leq C\int_{{\mathbb{R}}^3}|\xi_1||{\nabla}{{\bf{u}}}||{\nabla}^2{{\bf{B}}}|\,dx+C\int_{{\mathbb{R}}^3}|\sigma_1||{\nabla}{{\bf{u}}}||{\nabla}^2{{\bf{B}}}|\,dx \\ \leq & C\|\xi_1\|_{L^{\vec{p}}}\|{\nabla}{{\bf{u}}}\|_{L_{1}^{\frac{2p_1}{p_1-2}}L_{2}^{\frac{2p_2}{p_2-2}}L_{3}^{\frac{2p_3}{p_3-2}}}\|{\nabla}^2{{\bf{B}}}\|+C\|\sigma_1\|_{L^\infty}\|{\nabla}{{\bf{u}}}\|_{L^2}\|{\nabla}^2{{\bf{B}}}\|_{L^2}\\ \leq & C\|\xi_1\|_{L^{\vec{p}}}\|{\nabla}{{\bf{u}}}\|_{L^2}^{1-\sum\limits_{i = 1}^3\frac{1}{p_i}}\|{\nabla}^2{{\bf{u}}}\|_{L^2}^{\sum\limits_{i = 1}^{3}\frac{1}{p_i}}\|{\nabla}^2{{\bf{B}}}\|_{L^2}+C\|\sigma_1\|_{L^\infty}\|{\nabla}{{\bf{u}}}\|_{L^2}\|{\nabla}^2{{\bf{B}}}\|_{L^2}\\ \leq & C\|\xi_1\|_{L^{\vec{p}}}^2\|{\nabla}{{\bf{u}}}\|_{L^2}^{2-\sum\limits_{i = 1}^3\frac{2}{p_i}}\|{\nabla}^2{{\bf{u}}}\|_{L^2}^{\sum\limits_{i = 1}^{3}\frac{2}{p_i}}+\frac{\varepsilon}{2}\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2+C\|\sigma_1\|_{L^\infty}^2\|{\nabla}{{\bf{u}}}\|_{L^2}^2+\frac{\varepsilon}{2}\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2\\ \leq & C \|\xi_1\|_{L^{\vec{p}}}^{\frac{2}{1-\sum\limits_{i = 1}^3\frac{1}{p_i}}}\|{\nabla}{{\bf{u}}}\|_{L^2}^2+\varepsilon\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+C\|\sigma_1\|_{L^\infty}^2\|{\nabla}{{\bf{u}}}\|_{L^2}^2+\varepsilon \|{\nabla}^2{{\bf{B}}}\|_{L^2}^2. \end{align*}$

Recall the assumption that $\frac{2}{q}+\sum_{i = 1}^3\frac{1}{p_i} = 1$ , we have

$q = \frac{2}{1-\sum_{i = 1}^3\frac{1}{p_i}},$

hence,

$J_3\leq C(\|\xi_1\|_{L^{\vec{p}}}^q+\|\sigma_1\|_{L^\infty}^2)\|{\nabla}{{\bf{u}}}\|^2+\varepsilon (\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2).$

The estimates of $J_1, J_2, J_4, J_5$ and $J_6$ can be obtained in a similar way. Indeed, if we replace ${\nabla}^2{{\bf{B}}}$ by ${\nabla}^2{{\bf{u}}}$ in the estimate process for $J_3$ , we immediately get the estimate for $J_1$ :

$J_1\leq C(\|\xi_1\|_{L^{\vec{p}}}^q+\|\sigma_1\|_{L^\infty}^2)\|{\nabla}{{\bf{u}}}\|^2+\varepsilon \|{\nabla}^2{{\bf{u}}}\|_{L^2}^2.$

Replace ${\nabla}{{\bf{u}}}$ by ${\nabla}{{\bf{B}}}$ (and then ${\nabla}^2{{\bf{u}}}$ by ${\nabla}^2{{\bf{B}}}$ ) in the estimate process for $J_3$ and we can get the estimate for $J_2$ :

$J_2 \leq C(\|\xi_1\|_{L^{\vec{p}}}^q+\|\sigma_1\|_{L^\infty}^2)\|{\nabla}{{\bf{B}}}\|_{L^2}^2+\varepsilon \|{\nabla}^2{{\bf{B}}}\|_{L^2}^2.$

Replace ${{\bf{u}}}_h$ by ${{\bf{B}}}_h$ (and then $\xi_1$ by $\xi_2$ , $\sigma_1$ by $\sigma_2$ , $p_i$ by $k_i$ , $q$ by $l$ ) in the estimate process for $J_3$ and we can get the estimate for $J_5$ :

$J_5 \leq C(\|\xi_2\|_{L^{\vec{k}}}^l+\|\sigma_2\|_{L^\infty}^2)\|{\nabla}{{\bf{u}}}\|_{L^2}^2+\varepsilon (\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2).$

Now we estimate $J_4$ . By Lemma 2.4, Proposition 2.5 and Young's inequality, we have

$\begin{align*} &J_4 = C\int_{{\mathbb{R}}^3}|{{\bf{u}}}_h||{\nabla}{{\bf{B}}}||{\nabla}^2{{\bf{u}}}|\,dx \leq C\int_{{\mathbb{R}}^3}|\xi_1||{\nabla}{{\bf{B}}}||{\nabla}^2{{\bf{u}}}|\,dx+C\int_{{\mathbb{R}}^3}|\sigma_1||{\nabla}{{\bf{B}}}||{\nabla}^2{{\bf{u}}}|\,dx \\ \leq & C\|\xi_1\|_{L^{\vec{p}}}\|{\nabla}{{\bf{B}}}\|_{L_{1}^{\frac{2p_1}{p_1-2}}L_{2}^{\frac{2p_2}{p_2-2}}L_{3}^{\frac{2p_3}{p_3-2}}}\|{\nabla}^2{{\bf{u}}}\|_{L^2}+C\|\sigma_1\|_{L^\infty}\|{\nabla}{{\bf{B}}}\|_{L^2}\|{\nabla}^2{{\bf{u}}}\|_{L^2}\\ \leq & C\|\xi_1\|_{L^{\vec{p}}}\|{\nabla}{{\bf{B}}}\|_{L^2}^{1-\sum\limits_{i = 1}^3\frac{1}{p_i}}\|{\nabla}^2{{\bf{B}}}\|_{L^2}^{\sum\limits_{i = 1}^{3}\frac{1}{p_i}}\|{\nabla}^2{{\bf{u}}}\|_{L^2}+C\|\sigma_1\|_{L^\infty}\|{\nabla}{{\bf{B}}}\|_{L^2}\|{\nabla}^2{{\bf{u}}}\|_{L^2}\\ \leq & C\|\xi_1\|_{L^{\vec{p}}}^2\|{\nabla}{{\bf{B}}}\|_{L^2}^{2-\sum\limits_{i = 1}^3\frac{2}{p_i}}\|{\nabla}^2{{\bf{B}}}\|_{L^2}^{\sum\limits_{i = 1}^{3}\frac{2}{p_i}}+\frac{\varepsilon}{2}\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+C\|\sigma_1\|_{L^\infty}^2\|{\nabla}{{\bf{B}}}\|_{L^2}^2+\frac{\varepsilon}{2}\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2\\ \leq & C \|\xi_1\|_{L^{\vec{p}}}^{\frac{2}{1-\sum\limits_{i = 1}^3\frac{1}{p_i}}}\|{\nabla}{{\bf{B}}}\|_{L^2}^2+\varepsilon\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2+C\|\sigma_1\|_{L^\infty}^2\|{\nabla}{{\bf{B}}}\|_{L^2}^2+\varepsilon \|{\nabla}^2{{\bf{u}}}\|_{L^2}^2, \end{align*}$

since

$q = \frac{2}{1-\sum_{i = 1}^3\frac{1}{p_i}},$

we have

$J_4\leq C(\|\xi_1\|_{L^{\vec{p}}}^q+\|\sigma_1\|_{L^\infty}^2)\|{\nabla}{{\bf{B}}}\|^2+\varepsilon (\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2).$

Replace ${{\bf{u}}}_h$ by ${{\bf{B}}}_h$ (and then $\xi_1$ by $\xi_2$ , $\sigma_1$ by $\sigma_2$ , $p_i$ by $k_i$ , $q$ by $l$ ) in the estimate process for $J_4$ and we can get the estimate for $J_6$ :

$J_6 \leq C(\|\xi_2\|_{L^{\vec{k}}}^l+\|\sigma_2\|_{L^\infty}^2)\|{\nabla}{{\bf{B}}}\|_{L^2}^2+\varepsilon (\|{\nabla}^2{{\bf{u}}}\|_{L^2}^2+\|{\nabla}^2{{\bf{B}}}\|_{L^2}^2).$

Substituting the above estimates of $J_i$ , $i = 1, 2, \cdots, 6$ into (3.11), we obtain

$\begin{equation} \begin{aligned} &\frac{1}{2}\frac{d}{d t}(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})+\|{\nabla}^2 {\bf u}\|_{L^{2}}^{2}+\|{\nabla}^2 {\bf B}\|_{L^{2}}^{2}\\ &\leq C\left(\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\sigma_{1}\|_{L^{\infty}}^{2}\right)(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})\\ &+ C\left(\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{2}\|_{L^{\infty}}^{2}\right)(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})+6 \epsilon(\|\Delta {\bf u}\|_{L^{2}}^{2}+\|\Delta {\bf B}\|_{L^{2}}^{2}). \end{aligned} \end{equation}$

(3.12)

Using Lemma 2.6, we find

$\begin{equation} \begin{aligned} &\frac{d}{d t}(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})+C(\|{\nabla}^2{\bf u}\|_{L^{2}}^{2}+\|{\nabla}^2 {\bf B}\|_{L^{2}}^{2}) \\ &\leq C\left(\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^2\right)(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2})\\ & = C \frac{\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^2}{1+\ln (e+\|{\bf u}_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}})}(1+\ln (e+\|\bf u_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}}))(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}) \\ &\leq C\frac{\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^2}{1+\ln (e+\|{\bf u}_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}})}\\ &{\quad}{\quad}\times \left(1+\ln \left(e+\|{\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf u}\|_{L^{2}}^{2}+\|{\bf B}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}+C\right)\right)(\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}). \end{aligned} \end{equation}$

(3.13)

By (3.1), (3.2) and (3.13), we have

$\begin{aligned} & \frac{d}{d t}\left(1+\ln \left(e+\|{\bf u}\|_{L^{2}}^{2}+\|{{\bf{B}}}\|_{L^{2}}^{2}+\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}+C\right)\right)+C(\|{\nabla}^2 {\bf u}\|_{L^{2}}^{2}+\|{\nabla}^2 {\bf B}\|_{L^{2}}^{2}) \\ \leq & C\frac{\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^2}{1+\ln (e+\|\bf u_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}})}\left(1+\ln \left(e+\|{\bf u}\|_{L^{2}}^{2}+\|\nabla {\bf u}\|_{L^{2}}^{2}+\|{\bf B}\|_{L^{2}}^{2}+\|\nabla {\bf B}\|_{L^{2}}^{2}+C\right)\right). \end{aligned}$

Applying Gronwall's inequality, we get

$\begin{aligned} & \sup _{0 \leq t \leq T} \ln \left(e+\|{\bf u}\|_{L^{2}}^{2}+\|{\bf B}\|_{L^{2}}^{2}+\|\nabla {\bf u}\|_{L^{2}}^{2}+\|\nabla {{\bf{B}}}\|_{L^{2}}^{2}+C\right) \\ &\leq\left(1+\ln \left(e+\|{\bf u}_{0}\|_{L^{2}}^{2}+\|{\bf B}_{0}\|_{L^{2}}^{2}+\left\|\nabla {\bf u}_{0}\right\|_{L^{2}}^{2}+\left\|\nabla {\bf B}_{0}\right\|_{L^{2}}^{2}+C\right)\right) \\ &{\quad}\times \exp \left\{C \int_{0}^{T} \frac{\|\xi_{1}\|_{L^{\vec{p}}}^{q}+\|\xi_{2}\|_{L^{\vec{k}}}^{l}+\|\sigma_{1}\|_{L^{\infty}}^{2}+\|\sigma_{2}\|_{L^{\infty}}^2}{1+\ln (e+\|{\bf u}_{h}\|_{L^{\vec{s}}}+\|\bf B_{h}\|_{L^{\vec{r}}})} \,d t\right\}. \end{aligned}$

This gives

${{\bf{u}}},{{\bf{B}}}\in L^\infty(0,T;H^1({\mathbb{R}}^3))\cap L^2(0,T;H^2({\mathbb{R}}^3)).$

Now, we can prove Theorem 2.2. Since ${{\bf{u}}}_0, {{\bf{B}}}_0\in H^1({\mathbb{R}}^3)$ and ${{\rm{div}}}{{\bf{u}}}_0 = {{\rm{div}}}{{\bf{B}}}_0 = 0$ , the weak solution $({{\bf{u}}}, {{\bf{B}}})$ is strong and unique on $[0, T_1]$ for some $T_1 < T$ . The above a priori esitmate ensures that this strong solution can be extended beyond $T_1$ , which finally implies that $({{\bf{u}}}, {{\bf{B}}})$ is strong on $[0, T]$ and thus is regular up to $T$ . This completes the proof of Theorem 1.1.

4. Proof of Theorem 2.3

Let us introduce Elsässer's variables ${{\bf{W}}}^+$ and ${{\bf{W}}}^-$ :

${{{\bf{W}}}^+} = {\bf u}+{\bf B},{\quad} {{{\bf{W}}}^-} = {\bf u}-{\bf B},{\quad} {{{\bf{W}}}^+}(0) = {\bf u}_{0}+{\bf B}_{0}, {\quad}{{{\bf{W}}}^-}(0) = {\bf u}_{0}-{\bf B}_{0}$

and $\nabla P = \nabla\left(p+\frac{1}{2}|{\bf{B}}|^{2}\right)$ . Then, ${{{\bf{W}}}^+}, {{{\bf{W}}}^-}$ satisfies

$\begin{equation} \left\{\begin{array}{l} \partial_t{{{\bf{W}}}^+}-\Delta {{{\bf{W}}}^+}+({{{\bf{W}}}^-} \cdot \nabla) {{{\bf{W}}}^+}+\nabla P = 0,\\ \partial_t{{{\bf{W}}}^-}-\Delta {{{\bf{W}}}^-}+({{{\bf{W}}}^+} \cdot \nabla) {{{\bf{W}}}^-}+\nabla P = 0.\\ \end{array}\right. \end{equation}$

(4.1)

Multiplying the first equation of (4.1) by $|{{{\bf{W}}}^+}|^2 {{{\bf{W}}}^+}$ and the second equation of (4.1) by $|{{{\bf{W}}}^-}|^2 {{{\bf{W}}}^-}$ , integrating by parts and using the divergence free property of ${{\bf{W}}}^+$ and ${{\bf{W}}}^-$ , we conclude that

$\begin{equation} \begin{aligned} &\frac{1}{4} \frac{d}{d t}\|{{{\bf{W}}}^+}\|_{L^4}^4+\||{{{\bf{W}}}^+}||\nabla {{{\bf{W}}}^+}|\|_{L^2}^2+\frac{1}{2}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^2 = -\int_{\mathbb{R}^3} \nabla P \cdot|{{{\bf{W}}}^+}|^2 {{{\bf{W}}}^+} \,d x, \end{aligned} \end{equation}$

(4.2)

$\begin{equation} \begin{aligned} &\frac{1}{4} \frac{d}{d t}\|{{{\bf{W}}}^-}\|_{L^4}^4+\||{{{\bf{W}}}^-}||\nabla {{{\bf{W}}}^-}|\|_{L^2}^2+\frac{1}{2}\left\|\nabla|{{{\bf{W}}}^-}|^2\right\|_{L^2}^2 = -\int_{\mathbb{R}^3} \nabla P \cdot|{{{\bf{W}}}^-}|^2 {{{\bf{W}}}^-} \,d x.\\ \end{aligned} \end{equation}$

(4.3)

Combine (4.2) and (4.3) and the vector identity

$-\frac{{\bf{a}}}{|{\bf{a}}|}\cdot{\nabla}|{\bf{a}}| = |{\bf{a}}|{{\rm{div}}}\left(\frac{{\bf{a}}}{|{\bf{a}}|}\right)$

for ${\bf{a}} = {{\bf{W}}}^{\pm}$ , we have

$\begin{align} \label{eq4.4} &\frac{1}{4} \frac{d}{d t}(\|{{{\bf{W}}}^+}\|_{L^4}^4+\|{{{\bf{W}}}^-}\|_{L^4}^4)+\||{{{\bf{W}}}^+}||\nabla {{{\bf{W}}}^+}|\|_{L^2}^2+\||{{{\bf{W}}}^-}||\nabla {{{\bf{W}}}^-}|\|_{L^2}^2\\ &+\frac{1}{2}(\|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^2+\|\nabla|{{{\bf{W}}}^-}|^2\|_{L^2}^2)\\ = &-\int_{\mathbb{R}^3} \nabla P \cdot(|{{{\bf{W}}}^+}|^2 {{{\bf{W}}}^+}+|{{{\bf{W}}}^-}|^2 {{{\bf{W}}}^-}) \,d x = \int_{\mathbb{R}^3} P ({{{\bf{W}}}^+} \nabla|{{{\bf{W}}}^+}|^2+{{{\bf{W}}}^-}\nabla|{{{\bf{W}}}^-}| ^2) \,d x\\ \leq& C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^+}||{{{\bf{W}}}^+}|| \nabla|{{{\bf{W}}}^+}|| \,d x+C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^-}||{{{\bf{W}}}^-}|| \nabla|{{{\bf{W}}}^-}|| \,d x\\ \leq& C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^+}||{{{\bf{W}}}^+}|\left|\left(\frac{{{{\bf{W}}}^+}}{|{{{\bf{W}}}^+}|}| \nabla|{{{\bf{W}}}^+}|\right)\right| \,d x+C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^-}||{{{\bf{W}}}^-}|\left| \left(\frac{{{{\bf{W}}}^-}}{|{{{\bf{W}}}^-}|}\nabla|{{{\bf{W}}}^-}|\right)\right| \,d x\\ = &C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^+}||{{{\bf{W}}}^+}||{{{\bf{W}}}^+}|\left|{{\rm{div}}}\left(\frac{{{{\bf{W}}}^+}}{|{{{\bf{W}}}^+}|}\right)\right| \,d x+C\int_{\mathbb{R}^3} |P{{{\bf{W}}}^-}||{{{\bf{W}}}^-}||{{{\bf{W}}}^-}|\left|{{\rm{div}}}\left(\frac{{{{\bf{W}}}^-}}{|{{{\bf{W}}}^-}|}\right)\right| \,d x\\ = &K_{1}+K_{2}. \end{align}$

(4.4)

To estimate $K_1$ and $K_2$ , let us first establish an estimate of the pressure $P$ . Taking the divergence operator $\nabla \cdot$ on both sides of the first equation of (4.1), it follows that

$-\Delta P = {{\rm{div}}}(({{{\bf{W}}}^-}\cdot\nabla) {{{\bf{W}}}^+}) = {\partial}_j(W^-_i{\partial}_i W^+_j) = {\partial}_j{\partial}_i(W^-_i W^+_j).$

Here we used the divergence free property of ${{\bf{W}}}^-$ and the summation symbol $\sum_{i, j = 1}^3$ was omitted for convenience. As a consequence

$P = \mathcal{R}_i \mathcal{R}_j\left(W^+_iW^-_j\right),$

where $\mathcal{R}_i$ represents the classical three-dimensional Riesz transformations. By using the boundedness of Riesz transformations in $L^r(1 < r < \infty)$ space, we get

$\begin{equation} \|P\|_{L^r} \leq C\|{{{\bf{W}}}^+}\|_{L^{2 r}} \|{{{\bf{W}}}^-}\|_{L^{2 r}}. \end{equation}$

(4.5)

Using Hölder's inequality, Poincaré's inequality and (4.5), we obtain

$\begin{equation} \begin{aligned} &\|P {{{\bf{W}}}^+}\|_{L^2}^2 = \int_{{\mathbb{R}}^3} |P|^2|{{\bf{W}}}^+|^2\,dx \\ \leq &C\|P\|_{L^2}\|P\|_{L^3}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^6} \\ \leq & C\|{{{\bf{W}}}^+}\|_{L^4}\|{{{\bf{W}}}^-}\|_{L^4}\|{{{\bf{W}}}^+}\|_{L^6}\|{{{\bf{W}}}^-}\|_{L^6}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2} \\ \leq &C\|{{{\bf{W}}}^+}\|_{L^4}\|{{{\bf{W}}}^-}\|_{L^4}\|\nabla {{{\bf{W}}}^+}\|_{L^2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2} . \end{aligned} \end{equation}$

(4.6)

Similarly, we can get

$\begin{equation} \begin{aligned} \|P {{{\bf{W}}}^-}\|_{L^2}^2 \leq C\|{{{\bf{W}}}^+}\|_{L^4}\|{{{\bf{W}}}^-}\|_{L^4}\|\nabla {{{\bf{W}}}^+}\|_{L^2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}\left\|\nabla|{{{\bf{W}}}^-}|^2\right\|_{L^2} . \end{aligned} \end{equation}$

(4.7)

Now, we are in position of estimating $K_{1}$ and $K_{2}$ . For $K_1$ , recall the decompostion of $div\left(\frac{W^+}{|W^+|}\right)$ and Lemma 2.4, we have

$\begin{aligned} K_{1}&\leq C\|f_{1}\|_{L^{\vec{p}}}\| |{{{\bf{W}}}^+}|^2\|_{L_{1}^{\frac{2p_{1}}{p_{1}-2}}L_{2}^{\frac{2p_{2}}{p_{2}-2}}L_{3}^{\frac{2p_{3}}{p_{3}-2}}}\|P {{{\bf{W}}}^+}\|_{L^2}+C\|g_{1}\|_{L^{\infty}}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^2}\|P {{{\bf{W}}}^+}\|_{L^2}\\ &\leq C\|f_{1}\|_{L^{\vec{p}}}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^2}^{1-\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}\|P {{{\bf{W}}}^+}\|_{L^2}\\ &{\quad}+C\|g_{1}\|_{L^{\infty}}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^2}\|P {{{\bf{W}}}^+}\|_{L^2}\\ & = K_{1}^{'}+K_{1}^{''}. \end{aligned}$

Let's estimate $K_{1}^{'}$ first, by (4.6) we have

$\begin{align} K_{1}^{'}\leq& C\|f_{1}\|_{L^{\vec{p}}}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^2}^{1-\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}\\ &{\quad}\times\|{{{\bf{W}}}^+}\|_{L^4}^\frac{1}{2}\|{{{\bf{W}}}^-}\|_{L^4}^\frac{1}{2}\|\nabla {{{\bf{W}}}^+}\|_{L^2}^\frac{1}{2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^\frac{1}{2}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^\frac{1}{2}\\ \leq & C\|f_{1}\|_{L^{\vec{p}}}\left\|{{{\bf{W}}}^+}\right\|_{L^4}^{\frac{5}{2}-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^{\frac{1}{2}+\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}\|{{{\bf{W}}}^-}\|_{L^4}^\frac{1}{2}\|\nabla {{{\bf{W}}}^+}\|_{L^2}^\frac{1}{2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^\frac{1}{2}\\ \leq & \varepsilon \|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2} +C\|f_{1}\|_{L^{\vec{p}}}^{\frac{4}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}}\|{{{\bf{W}}}^+}\|_{L^4}^{\frac{4(\frac{5}{2}-2(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}))}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}}\|{{{\bf{W}}}^-}\|_{L^4}^{\frac{2}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}}\\ &{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}\times\|\nabla {{{\bf{W}}}^+}\|_{L^2}^{\frac{2}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^{\frac{2}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}}. \end{align}$

(4.8)

Let

$\alpha = \frac{4(\frac{5}{2}-2(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}))}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}\geq 3,{\quad} \beta = \frac{2}{3-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}\in \left(\frac{2}{3} , 1\right).$

It is easily checked that $\alpha+\beta = 4$ , therefore, by Young's inequality we have

$\begin{equation} \begin{aligned} \|{{{\bf{W}}}^+}\|_{L^4}^{\alpha}\|{{{\bf{W}}}^-}\|_{L^4}^{\beta} &\leq \frac{\alpha}{\alpha+\beta}\|{{{\bf{W}}}^+}\|_{L^4}^{\alpha\cdot\frac{\alpha+\beta}{\alpha}}+\frac{\beta}{\alpha+\beta}\|{{{\bf{W}}}^-}\|_{L^4}^{\beta\cdot\frac{\alpha+\beta}{\beta}}\\ &\leq \|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4}, \end{aligned} \end{equation}$

(4.9)

and

$\begin{equation} \begin{aligned} \|f_{1}\|_{L^{\vec{p}}}^{2\beta}\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{\beta}\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{\beta} &\leq(1-\beta)\|f_{1}\|_{L^{\vec{p}}}^\frac{2\beta}{1-\beta}+\beta(\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{\beta}\|\nabla{{{\bf{W}}}^-}^{\beta}\|_{L^2})^\frac{1}{\beta}\\ &\leq C(\|f_{1}\|_{L^{\vec{p}}}^{q}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2}), \end{aligned} \end{equation}$

(4.10)

where $q = \frac{2\beta}{1-\beta} = \frac{4}{1-2\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)}$ .

Concluding (4.8)–(4.10), we get the estimates for $K_1'$ :

$\begin{equation} \begin{aligned} K_{1}^{'}&\leq \varepsilon \|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2}+C(\|f_{1}\|_{L^{\vec{p}}}^{q}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4}). \end{aligned} \end{equation}$

(4.11)

The estimate of $K_{1}^{''}$ can be obtained in a similar way:

$\begin{equation} \begin{aligned} K_{1}^{''}&\leq C\|g_{1}\|_{L^{\infty}}\left\||{{{\bf{W}}}^+}|^2\right\|_{L^2}\|{{{\bf{W}}}^+}\|_{L^4}^\frac{1}{2}\|{{{\bf{W}}}^-}\|_{L^4}^\frac{1}{2}\|\nabla {{{\bf{W}}}^+}\|_{L^2}^\frac{1}{2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^\frac{1}{2}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^\frac{1}{2}\\ &\leq C\|g_{1}\|_{L^{\infty}}\|{{{\bf{W}}}^+}\|_{L^4}^\frac{5}{2}\|{{{\bf{W}}}^-}\|_{L^4}^\frac{1}{2}\|\nabla {{{\bf{W}}}^+}\|_{L^2}^\frac{1}{2}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^\frac{1}{2}\left\|\nabla|{{{\bf{W}}}^+}|^2\right\|_{L^2}^\frac{1}{2}\\ &\leq \varepsilon\|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2}+C\|g_{1}\|_{L^{\infty}}^\frac{4}{3}\|{{{\bf{W}}}^+}\|_{L^4}^\frac{10}{3}\|{{{\bf{W}}}^-}\|_{L^4}^\frac{2}{3}\|\nabla {{{\bf{W}}}^+}\|_{L^2}^\frac{2}{3}\|\nabla {{{\bf{W}}}^-}\|_{L^2}^\frac{2}{3}\\ &\leq \varepsilon \|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2}+C(\|g_{1}\|_{L^{\infty}}^{4}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4}). \end{aligned} \end{equation}$

(4.12)

Thus, we get

$\begin{equation} \begin{aligned} K_{1}&\leq 2\varepsilon \|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2}+C(\|f_{1}\|_{L^{\vec{p}}}^{q}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4})\\ &+C(\|g_{1}\|_{L^{\infty}}^{4}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4}). \end{aligned} \end{equation}$

(4.13)

Following a similar argument, we get the estimate for $K_{2}$ :

$\begin{equation} \begin{aligned} K_{2}&\leq 2\varepsilon \|\nabla|{{{\bf{W}}}^-}|^2\|_{L^2}^{2}+C(\|f_{2}\|_{L^{\vec{r}}}^{s}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4})\\ &+C(\|g_{2}\|_{L^{\infty}}^{4}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^-}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4}). \end{aligned} \end{equation}$

(4.14)

Substitute (4.13) and (4.14) into (4.4), we have

$\begin{aligned} &\frac{1}{4} \frac{d}{d t}(\|{{{\bf{W}}}^+}\|_{L^4}^4+\|{{{\bf{W}}}^-}\|_{L^4}^4)+\||{{{\bf{W}}}^+}||\nabla {{{\bf{W}}}^+}|\|_{L^2}^2+\||{{{\bf{W}}}^-}||\nabla {{{\bf{W}}}^-}|\|_{L^2}^2\\ &{\quad}+\frac{1}{2}(\|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^2+\|\nabla|{{{\bf{W}}}^-}|^2\|_{L^2}^2)\\ \leq& C(\|f_{1}\|_{L^{\vec{p}}}^{q}+\|f_{2}\|_{L^{\vec{r}}}^{s}+\sum\limits_{i = 1}^{2}\|g_{i}\|_{L^{\infty}}^{4}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2})(\|{{{\bf{W}}}^+}\|_{L^4}^{4}+\|{{{\bf{W}}}^-}\|_{L^4}^{4})\\ &{\quad}{\quad} +2\varepsilon( \|\nabla|{{{\bf{W}}}^+}|^2\|_{L^2}^{2}+\|\nabla|{{{\bf{W}}}^-}|^2\|_{L^2}^{2}).\\ \end{aligned}$

Choosing $\varepsilon$ small and applying Gronwall's inequality, we see that

$\begin{aligned} {}& \sup\limits_{0 \leq t \leq T}(\|{{{\bf{W}}}^+}\|_{L^4}^4+\|{{{\bf{W}}}^-}\|_{L^4}^4)\\ \leq &(\|{{{\bf{W}}}^+}(0)\|_{L^4}^4+\|{{{\bf{W}}}^-}(0)\|_{L^4}^4) \\ &\times\exp \left\{C\int_0^T(\|f_{1}\|_{L^{\vec{p}}}^{q}+\|f_{2}\|_{L^{\vec{r}}}^{s}+\sum\limits_{i = 1}^{2}\|g_{i}\|_{L^{\infty}}^{4}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2}+\|\nabla{{{\bf{W}}}^+}\|_{L^2}^{2})\,dt\right\}, \end{aligned}$

which implies that

${\bf u},{\bf B} \in L^{\infty}\left(0, T ; L^4\left(\mathbb{R}^3\right)\right) \subset L^8\left(0, T ; L^4\left(\mathbb{R}^3\right)\right),$

combining this with the Serrin-type regularity criteria for MHD system (see for example [4, Theorem 4]), we complete the proof of Theorem 2.3.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China Grant no. 12001069, the Natural Science Foundation of Chongqing Grant No. cstc2019jcyj-msxmX0214 and the Team Building Project for Graduate Tutors in Chongqing (yds223010).

Conflict of interest

The authors declare there is no conflicts of interest.

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On regularity criteria for MHD system in anisotropic Lebesgue spaces

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Abstract

1. Introduction

2. Preliminaries and main results

3. Proof of Theorem 2.2

4. Proof of Theorem 2.3

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Acknowledgments

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On regularity criteria for MHD system in anisotropic Lebesgue spaces

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2. Preliminaries and main results

3. Proof of Theorem 2.2

4. Proof of Theorem 2.3

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