A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field

doi:10.1016/j.na.2009.12.045

Nonlinear Analysis: Theory, Methods & Applications

Volume 72, Issues 9–10, 1 May 2010, Pages 3643-3648

https://doi.org/10.1016/j.na.2009.12.045 Get rights and content

Abstract

In this paper, we consider regularity criterion for solutions to the 3D viscous incompressible MHD equations in Morrey–Campanato spaces. It is proved that if the vorticity field $ω = \nabla \times u$ belongs to ${\dot{M}}_{2, \frac{3}{r}}$ for $0 < r \leq 1$ on $[0, T]$ , then the solution remains smooth on $[0, T]$ .

Introduction

In this paper, we consider the following 3D viscous incompressible MHD equations ${\begin{matrix} \partial_{t} u - Δ u + u \cdot \nabla u - b \cdot \nabla b + \nabla P = 0, \\ \partial_{t} b - Δ b + u \cdot \nabla b - b \cdot \nabla u = 0, \\ div u = div b = 0, \\ u (x, 0) = u_{0} (x), b (x, 0) = b_{0} (x), \end{matrix}$ where $u \in R^{3}$ is the velocity field, $b \in R^{3}$ is the magnetic field, $P (x, t)$ is a scalar pressure, and $u_{0} (x), b_{0} (x)$ with $div u_{0} = div b_{0} = 0$ in the sense of distribution are the initial velocity and magnetic fields.

It is well known [1] that the problem (1.1) is locally well posed for any given initial datum $u_{0}, b_{0} \in H^{s} (R^{3})$ , $s \geq 3$ . But whether this unique local solution can exist globally is a problem that presents a serious challenge. Some fundamental Serrin’s-type regularity criteria in terms of only the velocity was studied in [2], [3] independently. Recently, some improvement and extension was made based on these two basic papers. Part of them are listed here: Chen, Miao and Zhang [4] proved regularity by adding condition on $Δ_{j} (\nabla \times u)$ ; Zhou and Gala [5] proved regularity for $u$ and $\nabla u$ in the multiplier spaces; Wu [6] considered the velocity field being in the homogeneous Besov space; regularity was obtained by imposing condition on the pressure in [7] (a complete result was established in [8]); in [9] direction of vorticity field $\nabla \times u$ was discussed (see also [2]).

The purpose of this work is to improve the criterion on regularity of weak solutions under vorticity $ω = curl u \in L^{\frac{2}{2 - r}} ((0, T); {\dot{M}}_{2, \frac{3}{r}} (R^{3}))$ with $0 < r \leq 1$ . More precisely, we will prove

Theorem 1.1

Let $u_{0} \in H^{1} (R^{3})$ , $b_{0} \in L^{4} \cap L^{2} (R^{3})$ . Suppose that $(u, b)$ is a weak solution of MHD equations(1.1)on $[0, T)$ with $0 < T \leq \infty$ . If the vorticity field $ω = \nabla \times u$ satisfies $ω \in L^{\frac{2}{2 - r}} (0, T, {\dot{M}}_{2, \frac{3}{r}} (R^{3})) for 0 < r \leq 1,$ then the corresponding weak solution $(u, b)$ to(1.1)is smooth.

Section snippets

Preliminaries

Before stating our main result, we recall the definition and some properties of the space that we are going to use. These spaces play an important role in studying the regularity of solutions to partial differential equations (see e.g., [10], [11]).

Definition 2.1

For $1 < p \leq q \leq + \infty$ , the Morrey–Campanato space ${\dot{M}}_{p, q}$ is defined by : ${\dot{M}}_{p, q} = {f \in L_{l o c}^{p} (R^{3}) : {‖ f ‖}_{{\dot{M}}_{p, q}} = sup_{x \in R^{3}} sup_{R > 0} R^{3 / q - 3 / p} {‖ f ‖}_{L^{p} (B (x, R))} < \infty},$ where $B (x, R)$ denotes the ball of center $x$ with radius $R$ .

It is easy to verify that ${\dot{M}}_{p, q} (R^{3})$ is a Banach space under the

Proof of Theorem 1.1

In order to prove Theorem 1.1, first we should establish $b \in L^{\infty} (0, T; L^{4}) and | b | \nabla b \in L^{2} (0, T; L^{2}),$ if (1.2) holds.

First we consider the case of $0 < r < 1$ . We multiply the both sides of the second equation of (1.1) by $b {| b |}^{2}$ , the integration over $R^{3}$ to get, with the help of integration by parts, that $\frac{1}{4} \frac{d}{d t} {‖ b ‖}_{L^{4}}^{4} + \int_{R^{3}} {| b |}^{2} {| \nabla b |}^{2} + 2 {| b |}^{2} | \nabla | b {|^{2} |}^{2} d x \leq | \int_{R^{3}} (b \cdot \nabla u) {| b |}^{2} b d x | \leq {‖ {| b |}^{2} \nabla u ‖}_{L^{2}} {‖ b^{2} ‖}_{L^{2}} \leq C {‖ \nabla u ‖}_{{\dot{M}}_{2, \frac{3}{r}}} {‖ {| b |}^{2} ‖}_{{\dot{B}}_{2, 1}^{r}} {‖ b^{2} ‖}_{L^{2}} \leq C {‖ ω ‖}_{{\dot{M}}_{2, \frac{3}{r}}} {‖ {| b |}^{2} ‖}_{L^{2}}^{1 - r} {‖ \nabla {| b |}^{2} ‖}_{L^{2}}^{r} {‖ b ‖}_{L^{4}}^{2} \leq C {‖ ω ‖}_{{\dot{M}}_{2, \frac{3}{r}}} {‖ b ‖}_{L^{4}}^{2 (2 - r)} {‖ | b | \nabla b ‖}_{L^{2}}^{r} \leq C {({‖ ω ‖}_{{\dot{M}}_{2, \frac{3}{r}}}^{\frac{2}{2 - r}} {‖ b ‖}_{L^{4}}^{4})}^{\frac{2 - r}{2}} ({‖ | b | \nabla b ‖}_{L^{2}}$

Acknowledgements

The authors would like to thank the referee for his/her careful reading and constructive suggestions, which greatly improved the paper. This work is partially supported by the Program for New Century Excellent Talents in Universities (Grant No. NCET 07-0299), ZJNSF (Grant No. R6090109) and NSFC (Grant No. 10971197).

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A new regularity criterion for weak solutions to the viscous MHD equations in terms of the vorticity field

Abstract

Introduction

Section snippets

Preliminaries

Proof of Theorem 1.1

Acknowledgements

J. Differential Equations

Internat. J. Non-Linear Mech.

Ann. Inst. H. Poincar Anal. Non Linéaire

J. Math. Anal. Appl.

Some mathematical questions related to the MHD equations

Comm. Pure Appl. Math.

Remarks on regularities for the 3D MHD equations

Discrete Contin. Dyn. Syst.

On the regularity criterion of weak solution for the 3D viscous Magneto-Hydrodynamics equations

Comm. Math. Phys.