The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations

1.   Introduction

Let us consider the following Cauchy problem of the incompressible magneto-hydrodynamic (MHD) equations in three-spatial dimensions :

where $u$ is the velocity field, $b$ the magnetic field and $\pi$ the pressure, while $u_{0}, b_{0}$ are given initial data with $\nabla \cdot u_{0} = \nabla \cdot b_{0} = 0$ in the sense of distributions.

Due to the importance of both physics and mathematics, there is large literature on the well-posedness for weak solutions of magnetohydrodynamic equations. However, similar to the Navier-Stokes equations ($b = 0$), the question of global regularity of the weak solutions to (2) still one of the most challenging problems in the theory of PDE's (see for example [3,4,5,6,9,10,11,12,13] and the references therein). It is interesting to study the regularity of the weak solutions of (2) by imposing some growth sufficient conditions on the velocity or the pressure. In particular, the condition via only one directional derivative of the pressure was established in [2] and showed that the weak solution becomes regular if the pressure $\pi$ satisfies

Later on, Jia and Zhou [7] improve (3) as

Recently, some interesting logarithmical pressure regularity criteria of MHD equations are studied. In particular, Benbernou et al. [1] refined (4) by imposing the following regularity criterion

The aim of this paper is to establish the logaritmical regularity criterion in terms of the partial derivative of pressure in the framework of anisotropic Lebesgue space.

Throughout the rest of this paper, we endow the usual Lebesgue space $L^{p}( \mathbb{R}^{3})$ with the norm $\left\Vert \cdot \right\Vert _{L^{p}}$. We denote by $\partial _{i} = \frac{\partial }{\partial x_{i}}$ the partial derivative in the $x_{i}-$direction. Recall that the anisotropic Lebesgue space consists on all the total measurable real valued functions $h = h(x_{1}, x_{2}, x_{3})$ with finite norm

where $(i, j, k)$ belongs to the permutation group $S =$span$\left\{ 1, 2, 3\right\}$.

Before giving the main result, let us first recall the definition of weak solutions for MHD equations (2).

Definition 1.1 (weak solutions). Let $(u_{0}, b_{0})\in L^{2}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = \nabla \cdot b_{0} = 0$ in the sense of distribution and $T>0$. A pair vector field $\left( u(x, t), b(x, t)\right)$ is called a weak solution of (2) on $(0, T)$ if $(u, b)$ satisfies the following properties :

(ⅰ): $(u, b)\in L^{\infty }\left( \left( 0, T\right) ;L^{2}(\mathbb{R} ^{3})\right) \cap L^{2}\left( \left( 0, T\right) ;H^{1}(\mathbb{R} ^{3})\right) ;$

(ⅱ): $\nabla \cdot u = \nabla \cdot b = 0$ in the sense of distribution;

(ⅲ): $(u, b)$ verifies (2) in the sense of distribution.

(ⅳ): $(u, b)$ satisfies the energy inequality, that is,

for all $t\in \lbrack 0, T].$

Now, our result read as follows.

Theorem 1.2. Let $(u_{0}, b_{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R} ^{3})$ with $\nabla \cdot u_{0} = \nabla \cdot b_{0} = 0$ in $\mathbb{R}^{3}$. Suppose that $(u, b)$ is a weak solution of (2) in $(0, T)$. If the pressure $\pi$ satisfies the condition

where

then the weak solution $(u, b)$ becomes a regular solution on $(0, T]$.

This allows us to obtain the regularity criterion of weak solutions via only one directional derivative of the pressure. This extends and improve some known regularity criterion of weak solutions in term of one directional derivative, including the notable works of Jia and Zhou [7].

As an application of Theorem 1.2, we also obtain the following regularity criterion of weak solutions.

Corollary 1.3. Let $(u_{0}, b_{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ with $\nabla \cdot u_{0} = \nabla \cdot b_{0} = 0$ in the sense of distributions. Assume that $(u, b)$ is a weak solution of (2) in $(0, T)$. If the pressure satisfies the condition

where

then the weak solution $(u, b)$ becomes a regular solution on $(0, T]$.

In order to prove the main result, we need to recall the following lemma, which is proved in [9] (see also [8]).

Lemma 1.4. Let us assume that $r>1$ and $1<\gamma \leq \alpha <\infty$. Then for $f, g, \varphi \in C_{0}^{\infty }(\mathbb{R}^{3})$, we have the following estimate

where $0\leq \theta \leq 1$ satisfying

and

and $C$ is a constant independent of $f, g, \varphi$.

2.   Proof of Theorem 1.2

We are now ready to give the proof of Theorem 1.2. Clearly, in order to prove Theorem 1.2, it suffices to show that the assumption (5) ensures the following a priori estimate :

Proof. We first convert the MHD system equations into a symmetric form. Adding and substracting (2)$_{1}$ and (2)$_{2}$, we get $w^{+}$ and $w^{-}$ satisfy

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

Next, we establish some fundamental estimates between the pressure $\pi$ and $w^{\pm }$. Taking the divergence operator $\nabla \cdot$ on both sides of the first and second equations of (2.1) gives

and hence

where we have used the divergence free condition $\nabla \cdot w^{+} = \nabla \cdot w^{-} = 0$. Due to the boundedness of Riesz transform in Lebesgue space $L^{p}$ $(1<p<\infty )$, we have

Similarly, acting the operator $\nabla div$ on both sides of the first and second equations of (9), one shows that

together with Calderón-Zygmund inequality, implies that for any $1<p<\infty$,

or

By means of the local existence result, (2) with $(u_{0}, b_{0})\in L^{2}(\mathbb{R}^{3})\cap L^{4}(\mathbb{R}^{3})$ admit a unique $L^{4}-$ strong solution $(u, b)$ on a maximal time interval. For the notation simplicity, we may suppose that the maximal time interval is $[0, T)$. It is obvious that to prove regularity for $u$ and $b$, it is sufficient to prove it for $w^{+}$ and $w^{-}$. We shall show

and thus $\left\Vert u\right\Vert _{L^{4}}^{2}+\left\Vert b\right\Vert _{L^{4}}^{2}$ are uniformly bounded on $[0, T]$. This will lead to a contradiction to the estimates to be derived below. We now begin to follow this argument.

We multiply (9)$_{1}$ by $w^{+}\left\vert w^{+}\right\vert ^{2}$, (9)$_{2}$ by $w^{-}\left\vert w^{-}\right\vert ^{2}$ in $L^{2}( \mathbb{R}^{3})$, respectively and integrating by parts, we obtain

By Hölder's and Young's inequalities,

In the following, we establish the bound of the integral

on the right-hand side of (12).

We select that

in Lemma 1.4, then the selected $a$ and $b$ satisfy (7). To bound $A_{1}$ and $A_{2}$, we recall the

we can estimate $A_{1}$ as follows

where $\alpha , \beta , r$ and $\theta$ satisfy the following identities

According to the fact that $2\leq \lambda <3$, we choose $r = \frac{(4-\lambda )\alpha \gamma }{\gamma +\alpha \gamma -\alpha }$, then it follows from (13)$_{3}$ that $\beta = \frac{(3-\lambda )\gamma }{\gamma -1}$. Now, on the one hand, observe that

On the other hand, since

and since $\alpha \geq \gamma$, we get

But you know, $\lambda$ must be less than $3$, hence

which implies that $(3-\lambda )\alpha \gamma +(\alpha -\gamma )>0$. Gathering these estimates together, we obtain

and it is clear that

Now using Hölder inequality with exponents $\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]}$ and $\frac{2(3-\lambda )\alpha \gamma -\alpha (\lambda \gamma -3)}{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]}$, $A_{1}$ can be further estimated as

when $\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]} = 0$ $($i.e. $\alpha = \gamma = \frac{3}{\lambda })$ or

when $0<\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]}<1$ $($i.e. $\frac{3}{\lambda }<\gamma \leq \alpha < \frac{1}{\lambda -2})$ and $\beta = \frac{(3-\lambda )\gamma }{\gamma -1}$.

$A_{2}$ can be bounded exactly as $A_{1}$. Follwing the steps as in the bound of $A_{1}$, we have

when $\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]} = 0$ $($i.e. $\alpha = \gamma = \frac{3}{\lambda })$ or

when $0<\frac{\lambda \alpha \gamma -\alpha -2\gamma }{2[(3-\lambda )\alpha \gamma -\gamma +\alpha ]}<1$ $($i.e. $\frac{3}{\lambda }<\gamma \leq \alpha < \frac{1}{\lambda -2})$ and $\beta = \frac{(3-\lambda )\gamma }{\gamma -1}$.

Combining all the estimates from above, we find

Defining

and thanks to

inequality (14) implyes that

and hence

here

Thanks to $(w^{+}, w^{-})$ is a weak solution of the 3D MHD equations (2), that is

together with the interpolation inequality yields that

On the other hand, since

and

it is easy to see that

and consequently

Hence, one has

Applying the Gronwall inequality yields that

which together with (5) implies that

Hence, it follows from the triangle inequality and (18) that

and

Thus,

This completes the proof of Theorem 1.2.

Acknowledgments

This work was done while the second author was visiting the Catania University in Italy. He would like to thank the hospitality and support of the University, where this work was completed. This research is partially supported by Piano della Ricerca 2016-2018 - Linea di intervento 2: "Metodi variazionali ed equazioni differenziali". The fourth author wishes to thank the support of "RUDN University Program 5-100". The authors wish to express their thanks to the referee for his/her very careful reading of the paper, giving valuable comments and helpful suggestions.