1. Introduction

The MHD equation plays a significant role of mathematical model in fluid dynamics, which can be stated as follows :

$\left\{ \begin{array}{l} \partial _{t}u-\Delta u+u\cdot \nabla u+\nabla \pi -b\cdot \nabla b=0, \\ \partial _{t}b-\Delta b+u\cdot \nabla b-b\cdot \nabla u=0, \\ \nabla \cdot u=\nabla \cdot b=0, \\ u\left( x, 0\right) =u_{0}\left( x\right) , \text{  }b\left( x, 0\right) =b_{0}\left( x\right) . \end{array} \right.$

(1.1)

Here $u=u(x, t)\in \mathbb{R}^{3}$ is the velocity field, $\pi =\pi (x, t)\in \mathbb{R}$, $b=b(x, t)\in \mathbb{R}^{3}$ denote the velocity vector, scalar pressure and the magnetic field of the fluid, respectively, while $u_{0}(x)$ and $b_{0}(x)$ are given initial velocity and initial magnetic fields with $\nabla \cdot u_{0}=\nabla \cdot b_{0}=0$ in the sense of distribution.

In their work, Sermange and Temam ^[19] (see also Duvaut and Lions ^[6]) proved that the MHD equations admit at least one global weak solution for any divergence-free initial data $(u_{0}, b_{0})\in L^{2}(\mathbb{R}^{3})$ and it has a (unique) local strong solution, if additionally, $(u_{0}, b_{0})$ belongs to some Sobolev space $H^{s}(\mathbb{R}^{3})$ with $s\geq 3$. However, whether a local strong solution can exist globally, or equivalently, whether global weak solutions are smooth is an open and challenge problem.

There are many known mathematical results on the three-dimentional MHD equations (see ^{[4,5,10,14,20,21,22,25,26]} and the references therein). Realizing the dominant role played by the velocity field, He and Xin ^[10] were able to derive criteria in terms of the velocity field $u$ alone. In particular, a scaling invariant regularity criterion in terms of $u$ was established (also by Zhou ^[25] independently) which shows that a weak solution $(u, b)$ is smooth on a time interval $(0, T]$ if

$\nabla u\in L^{\alpha }(0, T;L^{\gamma }(\mathbb{R}^{3}))\text{ with } 1\leq \alpha ＜\infty , \text{ }3/2＜\gamma \leq \infty \text{ and }\frac{ 2}{\alpha }+\frac{3}{\gamma }=2.$

Moreover, the problem of so-called "regularity criteria via partial components" was shown in ^{[3,9,11,12,13,15,17,23,24,27]}.

Recently, Cao and Wu in ^[3] presented the regularity criteria on the derivatives of the pressure in one direction. More precisely, they proved that if

$\frac{\partial \pi }{\partial x_{3}}(x, t)\in L^{\alpha }\left( 0, T;L^{\gamma }(\mathbb{R}^{3})\right) \text{ with }\frac{2}{\alpha }+\frac{3}{ \gamma }\leq \frac{7}{4}\ \text{and }\frac{12}{7}\leq \gamma \leq \infty ,$

(1.2)

then $\left( u, b\right)$ is smooth on $\mathbb{R}^{3}\times \lbrack 0, T]$. Later, ^[13] and ^[24] improve condition (1.2) as:

$\frac{\partial \pi }{\partial x_{3}}(x, t)\in L^{\alpha }\left( 0, T;L^{\gamma }(\mathbb{R}^{3})\right) \text{ with }\frac{2}{\alpha }+\frac{3}{ \gamma }\leq 2\ \text{and }\frac{3}{2}\leq \gamma \leq \infty .$

(1.3)

Very recently, Benbernou et al. ^[2] extend (1.3) to the homogeneous Morrey-Campanato space $\overset{\cdot }{\mathcal{M}}_{2, \frac{3 }{r}}(\mathbb{R}^{3})$. to obtain the regularity of weak solutions. This space has been used successfully in the study of the uniqueness of weak solutions for the Navier-Stokes equations in ^[16] where it is pointed out that

$L^{\frac{3}{r}}\left( \mathbb{R}^{3}\right) \subset L^{\frac{3}{r}, \infty }\left( \mathbb{R}^{3}\right) \subset \overset{.}{\mathcal{M}}_{2, \frac{3}{r} }\left( \mathbb{R}^{3}\right) .$

The purpose of this manuscript is to establish a logarithmically improved regularity criterion in terms of the derivatives of the pressure in one direction of the systems (1.1). Our result can be stated as follows.

Theorem 1.1.(regularity criterion)Let $\left( u_{0}, b_{0}\right) \in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0}=\nabla \cdot b_{0}=0$. Suppose that $(u, b)$ is a weak solution to the MHD equations (1.1) in the time interval $[0, T)$ for some $0 ＜T＜\infty$. If the pressure $\pi (x, t)$ satisfies the condition :

$\int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert _{\overset{ \cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{2-r}}}{1+\ln (1+\left\Vert b\right\Vert _{L^{4}})}ds＜\infty \text{ for }0＜r＜1,$

then $\left( u, b\right)$ is a regular solution on $\mathbb{R}^{3}\times \lbrack 0, T]$.

Theorem 1.1 is also true for the 3-D incompressible Navier-Stokes equations, so it gives extensions for previous results in ^{[2,1,3,12,17,23]}. Definitions and basic properties of the Morrey-Campanato spaces can be find in ^[28] and the references therein. For concision, we omit them here.

Now we are in the position to prove Theorem 1.1.

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2. Proof of Theorem 1.1.

Throughout this paper, $C$ denotes a generic positive constant (generally large), it may be different from line to line. In order to prove regularity, we need to establish the $L^{4}$ bound of $(u, b)$ and the desired regularity then follows from the standard Serrin-type criteria on the 3D MHD equations.

Instead of considering the equations in the form (1.1.), we rewrite it in the following form as that in ^[7,8]:

$\left\{ \begin{array}{l} \partial _{t}w^{+}+w^{-}\cdot \nabla w^{+}=\Delta w^{+}-\nabla \pi , \\ \partial _{t}w^{-}+w^{+}\cdot \nabla w^{-}=\Delta w^{-}-\nabla \pi , \\ \nabla \cdot w^{+}=\nabla \cdot w^{-}=0, \\ w^{+}(x, 0)=u_{0}+b_{0}, \text{ }w^{-}(x, 0)=u_{0}-b_{0}, \end{array} \right.$

(2.1)

with $w^{\pm }:=u\pm b$.

First, taking the inner product of (2.1)$_{1}$ with $(0, 0, w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2})$, we have

$\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left\vert w_{3}^{+}\right\vert ^{4}dx+\int_{\mathbb{R}^{3}}(w^{-}\cdot \nabla )w_{3}^{+}.\left\vert w_{3}^{+}\right\vert ^{2}dx \\ =\int_{\mathbb{R}^{3}}\Delta w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2}dx-\int_{\mathbb{R}^{3}}\frac{\partial \pi }{\partial x_{3}} w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2}dx.$

Integrating by parts over $\mathbb{R}^{3}$ and using the divergence free property $\nabla \cdot w^{+}=0$ into account, we get

$\int_{\mathbb{R}^{3}}(w^{-}\cdot \nabla )w_{3}^{+}\cdot \left\vert w_{3}^{+}\right\vert ^{2}dx=0.$

For the second integral term, applying the integration by parts and the incompressible conditions again yield

$\int_{\mathbb{R}^{3}}\Delta w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2}dx=- \frac{3}{4}\int_{\mathbb{R}^{3}}\left\vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\vert ^{2}dx.$

We easily get

$\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left\vert w_{3}^{+}\right\vert ^{4}dx+\frac{3}{4}\int_{\mathbb{R}^{3}}\left\vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\vert ^{2}dx=-\int_{\mathbb{R}^{3}}\frac{ \partial \pi }{\partial x_{3}}w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2}dx. \label{eq1}$

(2.2)

Similarly, taking the inner product of the second equation of (2.1) with $(0, 0, w_{3}^{-}\left\vert w_{3}^{-}\right\vert ^{2})$, we obtain

$\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left\vert w_{3}^{-}\right\vert ^{4}dx+\frac{3}{4}\int_{\mathbb{R}^{3}}\left\vert \nabla \left\vert w_{3}^{-}\right\vert ^{2}\right\vert ^{2}dx=-\int_{\mathbb{R}^{3}}\frac{ \partial \pi }{\partial x_{3}}w_{3}^{-}\left\vert w_{3}^{-}\right\vert ^{2}dx. \label{eq2}$

(2.3)

Summing (2.2) and (2.3) together yields

$\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left( \left\vert w_{3}^{+}\right\vert ^{4}+\left\vert w_{3}^{-}\right\vert ^{4}\right) dx+ \frac{3}{4}\int_{\mathbb{R}^{3}}\left( \left\vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\vert ^{2}+\left\vert \nabla \left\vert w_{3}^{-}\right\vert ^{2}\right\vert ^{2}\right) dx \notag \\ =-\int_{\mathbb{R}^{3}}\frac{\partial \pi }{\partial x_{3}} w_{3}^{+}\left\vert w_{3}^{+}\right\vert ^{2}dx-\int_{\mathbb{R}^{3}}\frac{ \partial \pi }{\partial x_{3}}w_{3}^{-}\left\vert w_{3}^{-}\right\vert ^{2}dx \notag \\ =J_{1}+J_{2}. \label{eq5}$

(2.4)

In what follows, we will deal with each term on the right-hand side of (2.4) separately. We estimate $\left\Vert \frac{\partial \pi }{\partial x_{3}}\cdot \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}$ as follows :

$\left\Vert \frac{\partial \pi }{\partial x_{3}}\cdot \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}} \leq C\left\Vert \frac{ \partial \pi }{\partial x_{3}}\right\Vert _{\overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}\left\Vert \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{ \overset{.}{B}_{2, 1}^{r}} \\ \leq C\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{\overset {\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}\left\Vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}^{r}\left\Vert \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}^{1-r}.$

Here we have used the following inequality due to Machihara and Ozawa ^[18]

$\left\Vert f\right\Vert _{\overset{.}{B}_{2, 1}^{r}}\leq C\left\Vert f\right\Vert _{L^{2}}^{1-r}\left\Vert \nabla f\right\Vert _{L^{2}}^{r}\text{ for }0 ＜r＜1.$

Hence, it follows from the Hölder inequality and Young's inequality that

$\left\vert J_{1}\right\vert \leq \int_{\mathbb{R}^{3}}\left\vert \frac{ \partial \pi }{\partial x_{3}}\right\vert |w_{3}^{+}|^{3}dx \notag \\ \leq C\left\Vert \frac{\partial \pi }{\partial x_{3}}\cdot \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}\left\Vert w_{3}^{+}\right\Vert _{L^{2}} \notag \\ \leq C\left( \left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{ \overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}\right) ^{\frac{1-r}{2}}\left( \left\Vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}^{2}\right) ^{ \frac{r}{2}}\left( \left\Vert w^{+}\right\Vert _{L^{2}}^{2}\right) ^{\frac{1 }{2}} \notag \\ \leq C\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{\overset {\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\frac{1}{2}\left\Vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\Vert _{L^{2}}^{2}+C\left\Vert w^{+}\right\Vert _{L^{2}}^{2}, \label{eq3}$

(2.5)

Note that the weak solution $(u, b)\in L^{\infty }(0, T;L^{2}(\mathbb{R}^{3}))$, this leads to

$(w^{+}, w^{-})\in L^{\infty }(0, T;L^{2}(\mathbb{R}^{3})).$

Similarly, one can prove that

$\left\vert J_{2}\right\vert \leq C\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{\overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{ 1-r}}\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4}+\frac{1}{2}\left\Vert \nabla \left\vert w_{3}^{-}\right\vert ^{2}\right\Vert _{L^{2}}^{2}+C\left\Vert w^{-}\right\Vert _{L^{2}}^{2}. \label{eq4}$

(2.6)

Substituting (2.5) and (2.6) into (2.4), we obtain

$\frac{1}{4}\frac{d}{dt}\int_{\mathbb{R}^{3}}\left( \left\vert w_{3}^{+}\right\vert ^{4}+\left\vert w_{3}^{-}\right\vert ^{4}\right) dx+ \frac{1}{4}\int_{\mathbb{R}^{3}}\left( \left\vert \nabla \left\vert w_{3}^{+}\right\vert ^{2}\right\vert ^{2}+\left\vert \nabla \left\vert w_{3}^{-}\right\vert ^{2}\right\vert ^{2}\right) dx \notag \\ \leq C\left( \left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{ \overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}+1\right) \left( \left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4}\right) , \label{eq10}$

(2.7)

for all $0\leq t ＜T$. Setting

$J=\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{\overset{ \cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}\left( e+\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4}\right) .$

On the other hand, we see that

$1+\ln \left( 1+\left\Vert b\right\Vert _{L^{4}}\right) \leq 1+\ln \left( 1+\left\Vert b\right\Vert _{L^{4}}^{4}+\frac{9}{8}\right) \\ \leq 1+\ln \left( e+\left\Vert b\right\Vert _{L^{4}}^{4}\right) ,$

where we have used the following inequality

$x\leq x^{4}+\frac{9}{8}\text{ for all }x\geq 0.$

Consequently, $J$ can be estimated as follows:

$J =\frac{\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{ \overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b\right\Vert _{L^{4}})}(e+\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4})\left[1+\ln (1+\left\Vert b\right\Vert _{L^{4}})\right] \notag \\ \leq \frac{\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{ \overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b\right\Vert _{L^{4}})}(e+\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4})\left[1+\ln (e+\left\Vert b\right\Vert _{L^{4}}^{4})\right] \label{eq3.22} \\ \leq \frac{\left\Vert \frac{\partial \pi }{\partial x_{3}}\right\Vert _{ \overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b\right\Vert _{L^{4}})}(e+\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4})\left[1+\ln (e+\left\Vert w_{3}^{+}\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}\right\Vert _{L^{4}}^{4})\right] . \notag$

(2.8)

Inserting (2.8) into (2.7) and setting

$F(t)=e+\left\Vert w_{3}^{+}(t)\right\Vert _{L^{4}}^{4}+\left\Vert w_{3}^{-}(t)\right\Vert _{L^{4}}^{4},$

we obtain

$\frac{dF}{dt}\leq C\frac{\left\Vert \frac{\partial \pi }{\partial x_{3}} \right\Vert _{\overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r}}}^{\frac{2}{1-r}} }{1+\ln (1+\left\Vert b\right\Vert _{L^{4}})}(1+\ln F)F+CF.$

for all $t\in \left[0, T\right]$. Thank's to Gronwall inequality, we get

$F(t)\leq F(0)\exp \left( C\int_{0}^{t}\frac{\left\Vert \frac{\partial \pi }{ \partial x_{3}}(s)\right\Vert _{\overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r} }}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}(1+\ln F(s))ds\right) \exp (CT),$

which implies

$1+\ln F(t)\leq CT+\ln F(0)+C\int_{0}^{t}\frac{\left\Vert \frac{\partial \pi }{\partial x_{3}}(s)\right\Vert _{\overset{\cdot }{\mathcal{M}}_{2, \frac{3}{r }}}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}(1+\ln F(s))ds.$

Applying Gronwall's inequality again, one has

$\ln F(t)\leq c(u_{0}, b_{0}, T)\exp \left( C\int_{0}^{T}\frac{\left\Vert \frac{ \partial \pi }{\partial x_{3}}(s)\right\Vert _{\overset{\cdot }{\mathcal{M}} _{2, \frac{3}{r}}}^{\frac{2}{1-r}}}{1+\ln (1+\left\Vert b(s)\right\Vert _{L^{4}})}ds\right) ,$

which implies that

$\underset{0\leq t\leq T}{\sup }(\left\Vert w^{+}(., t)\right\Vert_{L^{4}}+\left\Vert w^{-}(., t)\right\Vert _{L^{4}}) ＜\infty \label{eq12.13}$

(2.9)

Hence, it follows from the triangle inequality and (2.9) that

$\underset{0\leq t\leq T}{\sup }\left\Vert u(., t)\right\Vert _{L^{4}} = \frac{1}{2}\underset{0\leq t\leq T}{\sup }\left\Vert (u+b)(., t)+(u-b)(., t)\right\Vert _{L^{4}} \\ \leq \frac{1}{2}\underset{0\leq t\leq T}{\sup }(\left\Vert (u+b)(., t)\right\Vert _{L^{4}}+\left\Vert (u-b)(., t)\right\Vert _{L4}) \\ \leq \frac{1}{2}\underset{0\leq t\leq T}{\sup }(\left\Vert w^{+}(., t)\right\Vert _{L^{4}}+\left\Vert w^{-}(., t)\right\Vert _{L^{4}}) ＜\infty$

and

$\underset{0\leq t\leq T}{\sup }\left\Vert b(., t)\right\Vert _{L^{4}} = \frac{1}{2}\underset{0\leq t\leq T}{\sup }\left\Vert (u+b)(., t)-(u-b)(., t)\right\Vert _{L^{4}} \\ \leq \frac{1}{2}\underset{0\leq t\leq T}{\sup }(\left\Vert (u+b)(., t)\right\Vert _{L^{4}}+\left\Vert (u-b)(., t)\right\Vert _{L^{4}}) \\ \leq \frac{1}{2}\underset{0\leq t\leq T}{\sup }(\left\Vert w^{+}(., t)\right\Vert _{L^{4}}+\left\Vert w^{-}(., t)\right\Vert _{L^{4}})＜\infty .$

Thus,

$\underset{0\leq t\leq T}{\sup }(\left\Vert u(., t)\right\Vert_{L^{4}}+\left\Vert b(., t)\right\Vert _{L^{4}})＜\infty .$

(2.10)

This completes the proof of Theorem 1.1.

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Acknowledgments

Part of the work was carried out while the first author was long term visitor at University of Catania. The hospitality and support of Catania University are graciously acknowledged.

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Conflict of Interest

All authors declare no conflicts of interest in this paper.

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