Skip to main content

A regularity criterion for the generalized Hall-MHD system

Abstract

This paper proves a regularity criterion for the 3D generalized Hall-MHD system.

Introduction

In this paper, we consider the following 3D generalized Hall-MHD system:

$$\begin{aligned}& \partial_{t}u+u\cdot\nabla u+\nabla \biggl(\pi+ \frac{1}{2}|b|^{2} \biggr)+(-\Delta)^{\alpha}u=b\cdot\nabla b, \end{aligned}$$
(1.1)
$$\begin{aligned}& \partial_{t}b+u\cdot\nabla b-b\cdot\nabla u+(- \Delta)^{\beta}b+\operatorname {rot}(\operatorname {rot}b\times b)=0, \end{aligned}$$
(1.2)
$$\begin{aligned}& \operatorname {div}u=\operatorname {div}b=0, \end{aligned}$$
(1.3)
$$\begin{aligned}& (u,b) (\cdot,0)=(u_{0},b_{0}) \quad\mbox{in } {\mathbb {R}^{3}}. \end{aligned}$$
(1.4)

Here u, π, and b denote the velocity, pressure, and magnetic field of the fluid, respectively. \(0<\alpha,\beta\) are two constants. The fractional Laplace operator \((-\Delta)^{\alpha}\) is defined through the Fourier transform, namely, \(\widehat{(-\Delta)^{\alpha}f}(\xi )=|\xi|^{2\alpha}\hat{f}(\xi)\).

The applications of the Hall-MHD system cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars, and the geo-dynamo.

When the Hall effect term \(\operatorname {rot}(\operatorname {rot}b\times b)\) is neglected, the system (1.1)-(1.4) reduces to the well-known generalized MHD system, which has received attention in many studies [16].

When \(\alpha=\beta=1\), the system (1.1)-(1.4) reduces to the well-known Hall-MHD system, which has received many studies [711]. Reference [7] gave a derivation of the isentropic Hall-MHD system from a two-fluid Euler-Maxwell system. Chae-Degond-Liu [8] proved the local existence of smooth solutions. Chae and Schonbek [9] showed the time-decay and some regularity criteria were proved in [1012]. Some other relevant results about Hall-MHD equations can be found in [1317].

The local well-posedness is established in Wan and Zhou [18] when \(0<\alpha\leq1\) and \(\frac{1}{2}<\beta\leq1\).

When \(\frac{3}{4}\leq\alpha<\frac{5}{4}\) and \(1\leq\beta<\frac{7}{4}\), Jiang and Zhu [19] prove the following regularity criteria:

$$\nabla b\in L^{t}\bigl(0,T;L^{s}\bigr)\quad\mbox{with } \frac{2\beta}{t}+\frac {3}{s}\leq2\beta-1, \frac{3}{2\beta-1}< s\leq\infty, $$

and one of the following two conditions:

$$u\in L^{p}\bigl(0,T;L^{q}\bigr)\quad\mbox{with } \frac{2\alpha}{p}+\frac{3}{q}\leq 2\alpha-1, \frac{3}{2\alpha-1}< q\leq \frac{6\alpha}{2\alpha-1}, $$

or

$$\Lambda^{\alpha}u\in L^{p}\bigl(0,T;L^{q}\bigr) \quad\mbox{with } \frac{2\alpha }{p}+\frac{3}{q}\leq3\alpha-1, \frac{3}{3\alpha-1}< q\leq\frac {6\alpha}{3\alpha-1}. $$

When \(1\leq\alpha<\frac{5}{4}\) and \(1\leq\beta<\frac{7}{4}\), Ye [20] showed the following regularity criterion:

$$u\in L^{p}\bigl(0,T;L^{q}\bigr)\quad\mbox{and}\quad\nabla b\in L^{\ell}\bigl(0,T;L^{k}\bigr), $$

where p, q, , and k satisfy the relation

$$\frac{3}{q}+\frac{2\alpha}{p}\leq2\alpha-1, \qquad \frac{3}{q}+ \frac{2\beta }{p}\leq2\beta-1, \qquad \frac{3}{k}+\frac{2\beta}{\ell}\leq2\beta-1, $$

and

$$\max \biggl(\frac{3}{2\alpha-1},\frac{3}{2\beta-1} \biggr)< q\leq \infty,\qquad \frac{3}{2\beta-1}< k\leq\infty. $$

The aim of this paper is to refine the results in [19, 20] as follows.

Theorem 1.1

Let \(\frac{3}{4}\leq\alpha<\frac{5}{4}\) and \(\frac{3}{4}\leq \beta<\frac{7}{4}\) and \(u_{0},b_{0}\in H^{2}\) with \(\operatorname {div}u_{0}=\operatorname {div}b_{0}=0\) in \({\mathbb {R}^{3}}\). If u and b satisfy

$$ \begin{aligned} & \nabla u\in L^{\frac{2\alpha}{2\alpha-\gamma_{1}}}\bigl(0,T; \dot{B}_{\infty ,\infty}^{-\gamma_{1}}\bigr),\\ & \nabla b\in L^{\frac{2\beta}{2\beta -1-\gamma_{2}}}\bigl(0,T;\dot{B}_{\infty,\infty}^{-\gamma_{2}} \bigr)\quad \textit{with } 0< \gamma_{1}< 2\alpha\textit{ and }0< \gamma_{2}< 2 \beta-1, \end{aligned} $$
(1.5)

with \(0< T<\infty\), then the solution \((u,b)\) can be extended beyond T.

In the following proof, we will use the following bilinear products and commutator estimates due to Kato-Ponce [21]:

$$\begin{aligned}& \bigl\Vert \Lambda^{s}(fg)\bigr\Vert _{L^{p}}\leq C\bigl( \bigl\Vert \Lambda^{s}f\bigr\Vert _{L^{p_{1}}}\Vert g\Vert _{L^{q_{1}}}+\Vert f\Vert _{L^{p_{2}}}\bigl\Vert \Lambda^{s}g\bigr\Vert _{L^{q_{2}}}\bigr), \end{aligned}$$
(1.6)
$$\begin{aligned}& \bigl\Vert \Lambda^{s}(fg)-f\Lambda^{s}g\bigr\Vert _{L^{p}}\leq C\bigl(\Vert \nabla f\Vert _{L^{p_{1}}}\bigl\Vert \Lambda^{s-1}g\bigr\Vert _{L^{q_{1}}}+\bigl\Vert \Lambda^{s}f\bigr\Vert _{L^{p_{2}}}\Vert g\Vert _{L^{q_{2}}} \bigr), \end{aligned}$$
(1.7)

with \(s>0\), \(\Lambda:=(-\Delta)^{\frac{1}{2}}\), and \(\frac{1}{p}=\frac {1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}\).

We will also use the improved Gagliardo-Nirenberg inequalities [2224]:

$$\begin{aligned}& \Vert \nabla u\Vert _{L^{3}}^{3}\leq C\Vert \nabla u \Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{1}}}\Vert \nabla u\Vert _{\dot{H}^{\frac{\gamma_{1}}{2}}}^{2}, \end{aligned}$$
(1.8)
$$\begin{aligned}& \Vert \nabla b\Vert _{L^{3}}^{3}\leq C\Vert \nabla b \Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{2}}}\Vert \nabla b\Vert _{\dot{H}^{\frac{\gamma_{2}}{2}}}^{2}, \end{aligned}$$
(1.9)
$$\begin{aligned}& \Vert \nabla b\Vert _{L^{p_{3}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{2}}}^{1-\theta_{1}}\Vert \nabla b\Vert _{\dot{H}^{s}}^{\theta_{1}}, \end{aligned}$$
(1.10)
$$\begin{aligned}& \bigl\Vert \Lambda^{2-\beta}b\bigr\Vert _{L^{q_{3}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty ,\infty}^{-\gamma_{2}}}^{\theta_{1}}\Vert \nabla b\Vert _{\dot{H}^{s}}^{\theta _{1}}, \end{aligned}$$
(1.11)

with \(\theta_{1}:=\frac{\gamma_{2}}{s+\gamma_{2}}\), \(s:=1+\gamma_{2}-\beta\), \(p_{3}:=\frac{2}{\theta_{1}}\) and \(q_{3}:=\frac{2}{1-\theta_{1}}\)

$$\begin{aligned}& \Vert \nabla b\Vert _{L^{p_{4}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{2}}}^{1-\theta_{2}}\Vert \Delta b\Vert _{\dot{H}^{s}}^{\theta_{2}}, \end{aligned}$$
(1.12)
$$\begin{aligned}& \bigl\Vert \Lambda^{3-\beta}b\bigr\Vert _{L^{q_{4}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty ,\infty}^{-\gamma_{2}}}^{\theta_{2}}\Vert \Delta b\Vert _{\dot{H}^{s}}^{1-\theta _{2}}, \end{aligned}$$
(1.13)

with \(\theta_{2}:=\frac{\gamma_{2}}{1+s+\gamma_{2}}\), \(p_{4}:=\frac{2}{\theta _{2}}\), and \(q_{4}:=\frac{2}{1-\theta_{2}}\). We have

$$\begin{aligned}& \bigl\Vert \Lambda^{2-\beta}b\bigr\Vert _{L^{p_{5}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty ,\infty}^{-\gamma_{2}}}^{1-\theta_{3}}\Vert \Delta b\Vert _{\dot{H}^{s}}^{\theta _{3}}, \end{aligned}$$
(1.14)
$$\begin{aligned}& \Vert \Delta b\Vert _{L^{q_{5}}}\leq C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{2}}}^{\theta_{3}}\Vert \Delta b\Vert _{\dot{H}^{s}}^{1-\theta_{3}}, \end{aligned}$$
(1.15)

with \(\theta_{3}:=\frac{s}{1+s+\gamma_{2}}\), \(p_{5}:=\frac{2}{\theta_{3}}\), and \(q_{5}:=\frac{2}{1-\theta_{3}}\).

Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1, we only need to prove a priori estimates.

First, testing (1.1) by u and using (1.3), we see that

$$\frac{1}{2}\frac{d}{dt} \int|u|^{2}\,dx+ \int\bigl|\Lambda^{\alpha}u\bigr|^{2}\,dx= \int (b\cdot\nabla)b\cdot u \,dx. $$

Testing (1.2) by b and using (1.3), we find that

$$\frac{1}{2}\frac{d}{dt} \int|b|^{2}\,dx+ \int\bigl|\Lambda^{\beta}b\bigr|^{2} \,dx= \int (b\cdot\nabla)u\cdot b \,dx. $$

Summing up the above two equations, we get the well-known energy inequality

$$ \frac{1}{2} \int\bigl(|u|^{2}+|b|^{2}\bigr)\,dx+ \int_{0}^{T}\!\! \int\bigl(\bigl|\Lambda^{\alpha}u\bigr|^{2}+\bigl|\Lambda^{\beta}b\bigr|^{2}\bigr)\,dx \,dt\leq\frac{1}{2} \int \bigl(|u_{0}|^{2}+|b_{0}|^{2} \bigr)\,dx. $$
(2.1)

Testing (1.1) by \(-\Delta u\) and using (1.3), we infer that

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int|\nabla u|^{2} \,dx+ \int\bigl|\Lambda^{1+\alpha }u\bigr|^{2} \,dx \\& \quad= \int u\cdot\nabla u\cdot\Delta u \,dx- \int b\cdot\nabla b\cdot\Delta u \,dx \\& \quad=-\sum_{i,j} \int\partial_{j}u_{i}\,\partial_{i}u\, \partial_{j}u \,dx+\sum_{i,j} \int\partial_{j}b_{i}\,\partial_{i}b \,\partial_{j}u \,dx+\sum_{i,j} \int b_{i}\partial_{i} \partial_{j}b\, \partial_{j}u \,dx \\& \quad\leq C\Vert \nabla u\Vert _{L^{3}}^{3}+C\Vert \nabla b \Vert _{L^{3}}^{3}+\sum_{i,j} \int b_{i}\partial_{i}\partial_{j}b\, \partial_{j}u \,dx=:I_{1}+I_{2}+I_{3}. \end{aligned}$$
(2.2)

Testing (1.2) by \(-\Delta b\) and using (1.3), we deduce that

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int|\nabla b|^{2}\,dx+ \int\bigl|\Lambda^{1+\beta }b\bigr|^{2} \,dx \\ & \quad= \int u\cdot\nabla b\cdot\Delta b \,dx- \int b\cdot\nabla u\cdot \Delta b \,dx+ \int(\operatorname {rot}b\times b)\Delta \operatorname {rot}b \,dx \\ & \quad=-\sum_{i,j} \int\partial_{j}u_{i}\,\partial_{i}b\, \partial_{j}b \,dx+\sum_{i,j} \int\partial_{j}b_{i}\,\partial_{i}u\, \partial_{j}b \,dx \\ & \qquad{}+\sum_{i,j} \int b_{i}\partial_{i}\partial_{j}u\, \partial_{j}b \,dx-\sum_{i} \int(\operatorname {rot}b\times\partial_{i}b)\,\partial_{i}\operatorname {rot}b \,dx \\ & \quad\leq C\Vert \nabla u\Vert _{L^{3}}^{3}+C\Vert \nabla b \Vert _{L^{3}}^{3}+\sum_{i,j} \int b_{i}\partial_{i}\partial_{j}u\, \partial_{j}b \,dx-\sum_{i} \int(\operatorname {rot}b\times\partial_{i}b)\,\partial_{i}\operatorname {rot}b \,dx \\ & \quad=:I_{1}+I_{2}+I_{4}+I_{5}. \end{aligned}$$
(2.3)

Summing up (2.2) and (2.3), using (1.6), (1.8), (1.9), (1.10), (1.11), and \(I_{3}+I_{4}=0\), we derive

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int\bigl(|\nabla u|^{2}+|\nabla b|^{2}\bigr)\,dx+ \int \bigl(\bigl|\Lambda^{1+\alpha}u\bigr|^{2}+\bigl|\Lambda^{1+\beta}b\bigr|^{2} \bigr)\,dx \\ & \quad\leq C\Vert \nabla u\Vert _{L^{3}}^{3}+C\Vert \nabla b \Vert _{L^{3}}^{3}-\sum_{i} \int \Lambda^{1-\beta}(\operatorname {rot}b\times\partial_{i}b)\cdot \Lambda^{\beta -1}\,\partial_{i}\operatorname {rot}b \,dx \\ & \quad\leq C\Vert \nabla u\Vert _{L^{3}}^{3}+C\Vert \nabla b \Vert _{L^{3}}^{3}+C\Vert \nabla b\Vert _{L^{p_{3}}} \bigl\Vert \Lambda^{2-\beta}b\bigr\Vert _{L^{q_{3}}}\bigl\Vert \Lambda^{1+\beta}b\bigr\Vert _{L^{2}} \\ & \quad\leq C\Vert \nabla u\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{1}}}\Vert \nabla u\Vert _{\dot{H}^{\frac{\gamma_{1}}{2}}}^{2}+C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{2}}}\Vert \nabla b\Vert _{\dot{H}^{\frac{\gamma_{2}}{2}}}^{2}+C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}\Vert \nabla b\Vert _{\dot{H}^{s}}\bigl\Vert \Lambda^{1+\beta}b\bigr\Vert _{L^{2}} \\ & \quad\leq C\Vert \nabla u\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{1}}}\Vert \nabla u\Vert _{L^{2}}^{2 (1-\frac{\gamma_{1}}{2\alpha} )}\bigl\Vert \Lambda ^{1+\alpha}u\bigl\Vert _{L^{2}}^{2\cdot\frac{\gamma_{1}}{2\alpha}}+C\bigr\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}} \Vert \nabla b\bigr\Vert _{L^{2}}^{2 (1-\frac{\gamma_{2}}{2\beta} )}\bigl\Vert \Lambda^{1+\beta}b\bigr\Vert _{L^{2}}^{2\cdot\frac{\gamma_{2}}{2\beta}} \\ & \qquad{}+C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}\Vert \nabla b\Vert _{L^{2}}^{1-\frac{s}{\beta}}\bigl\Vert \Lambda^{1+\beta}b\bigr\Vert _{L^{2}}^{1+\frac {s}{\beta}} \\ & \quad\leq \frac{1}{2}\bigl\Vert \Lambda^{1+\alpha}u\bigr\Vert _{L^{2}}^{2}+\frac{1}{2}\bigl\Vert \Lambda ^{1+\beta}b\bigr\Vert _{L^{2}}^{2}+C\Vert \nabla u \Vert _{\dot{B}_{\infty,\infty}^{-\gamma _{1}}}^{\frac{2\alpha}{2\alpha-\gamma_{1}}} \Vert \nabla u\Vert _{L^{2}}^{2} \\ & \qquad{}+C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}^{\frac{2\beta }{2\beta-\gamma_{2}}}\Vert \nabla b\Vert _{L^{2}}^{2}+C \Vert \nabla b\Vert _{\dot{B}_{\infty ,\infty}^{-\gamma_{2}}}^{\frac{2\beta}{2\beta-1-\gamma_{2}}}\Vert \nabla b\Vert _{L^{2}}^{2}, \end{aligned}$$

which gives

$$ \bigl\Vert (u,b)\bigr\Vert _{L^{\infty}(0,T;H^{1})}+\Vert u\Vert _{L^{2}(0,T;H^{1+\alpha})}+ \Vert b\Vert _{L^{2}(0,T;H^{1+\beta})}\leq C. $$
(2.4)

In the following proofs, we will use the following Sobolev embedding theorem:

$$ \begin{aligned} & \Vert \nabla u\Vert _{L^{4}}\leq C\Vert u\Vert _{H^{1+\alpha}},\qquad \Vert \Delta u\Vert _{L^{4}}\leq C\Vert u\Vert _{H^{2+\alpha}}\leq C\bigl\Vert \Lambda^{2+\alpha}u\bigr\Vert _{L^{2}}+C,\\ & \Vert \nabla b\Vert _{L^{4}}\leq C\Vert b\Vert _{H^{1+\beta}}, \qquad \Vert \Delta b\Vert _{L^{4}}\leq C\bigl\Vert \Lambda^{2+\beta}b\bigr\Vert _{L^{2}}+C. \end{aligned} $$
(2.5)

Taking Δ to (1.1), testing by Δu and using (1.3), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int|\Delta u|^{2} \,dx+ \int\bigl|\Lambda^{2+\alpha }u\bigr|^{2} \,dx \\& \quad =- \int\bigl(\Delta(u\cdot\nabla u)-u\cdot\nabla\Delta u\bigr)\Delta u \,dx + \int\bigl(\Delta(b\cdot\nabla b)-b\cdot\nabla\Delta b\bigr)\Delta u \,dx \\& \qquad{}+ \int b\cdot\nabla\Delta b\cdot\Delta u \,dx \\& \quad =:I_{6}+I_{7}+I_{8}. \end{aligned}$$
(2.6)

Applying Δ to (1.2), testing by Δb and using (1.3), we have

$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int|\Delta b|^{2} \,dx+ \int\bigl|\Lambda^{2+\beta }b\bigr|^{2} \,dx \\& \quad =- \int\bigl(\Delta(u\cdot\nabla b)-u\cdot\nabla\Delta b\bigr)\Delta b \,dx + \int\bigl(\Delta(b\cdot\nabla u)-b\cdot\nabla\Delta u\bigr)\Delta b \,dx \\& \qquad{}+ \int b\cdot\nabla\Delta u\cdot\Delta b \,dx- \int\Delta(\operatorname {rot}b\times b)\cdot\Delta \operatorname {rot}b \,dx \\& \quad=:I_{9}+I_{10}+I_{11}+I_{12}. \end{aligned}$$
(2.7)

Note that \(I_{8}+I_{11}=0\).

Using (1.7), (2.4), and (2.5), we bound \(I_{6}+I_{7}+I_{9}+I_{10}\) as follows:

$$\begin{aligned}& I_{6}+I_{7}+I_{9}+I_{10}\\& \quad\leq C \Vert \nabla u\Vert _{L^{4}}\Vert \Delta u\Vert _{L^{4}} \Vert \Delta u\Vert _{L^{2}} +C\Vert \nabla b\Vert _{L^{4}}\Vert \Delta b\Vert _{L^{4}}\Vert \Delta u\Vert _{L^{2}}\\& \qquad{}+C\bigl(\Vert \nabla b \Vert _{L^{4}}\Vert \Delta u\Vert _{L^{4}}+\Vert \nabla u \Vert _{L^{4}}\Vert \Delta b\Vert _{L^{4}}\bigr)\Vert \Delta b\Vert _{L^{2}} \\& \quad\leq C\Vert u\Vert _{H^{1+\alpha}} \Vert u\Vert _{H^{2+\alpha}} \Vert \Delta u\Vert _{L^{2}}+C\Vert b\Vert _{H^{1+\beta}} \Vert b\Vert _{H^{2+\beta}} \Vert \Delta u\Vert _{L^{2}} \\& \qquad{}+C\bigl(\Vert b\Vert _{H^{1+\beta}} \Vert u\Vert _{H^{2+\alpha}}+ \Vert u\Vert _{H^{1+\alpha}} \Vert b\Vert _{H^{2+\beta}}\bigr)\Vert \Delta b\Vert _{L^{2}} \\& \quad\leq \frac{1}{8}\Vert u\Vert _{H^{2+\alpha}}^{2}+ \frac{1}{8}\Vert b\Vert _{H^{2+\beta }}^{2}+C\bigl(\Vert u \Vert _{H^{1+\alpha}}^{2}+\Vert b\Vert _{H^{1+\beta}}^{2} \bigr) \bigl(\Vert \Delta u\Vert _{L^{2}}^{2}+\Vert \Delta b\Vert _{L^{2}}^{2}\bigr). \end{aligned}$$

Using (1.6), (1.12), (1.13), (1.14), and (1.15), we bound \(I_{12}\) as follows:

$$\begin{aligned} I_{12} =&- \int(\operatorname {rot}b\times\Delta b)\cdot\Delta \operatorname {rot}b \,dx-2\sum_{i} \int(\partial_{i}\operatorname {rot}b\times\partial_{i}b)\Delta \operatorname {rot}b \,dx \\ =&- \int\Lambda^{1-\beta}(\operatorname {rot}b\times\Delta b)\cdot\Lambda^{\beta -1} \Delta \operatorname {rot}b \,dx\\ &{}-2\sum_{i} \int\Lambda^{1-\beta}(\partial _{i}\operatorname {rot}b\times \partial_{i}b)\cdot\Lambda^{\beta-1}\Delta \operatorname {rot}b \,dx \\ \leq&C\bigl(\Vert \nabla b\Vert _{L^{p_{4}}}\bigl\Vert \Lambda^{3-\beta}b\bigr\Vert _{L^{q_{4}}}+\bigl\Vert \Lambda^{2-\beta}b\bigr\Vert _{L^{p_{5}}}\Vert \Delta b\Vert _{L^{q_{5}}}\bigr)\bigl\Vert \Lambda ^{2+\beta}b\bigr\Vert _{L^{2}} \\ \leq&C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}\Vert \Delta b\Vert _{\dot{H}^{s}}\bigl\Vert \Lambda^{2+\beta}b\bigr\Vert _{L^{2}} \\ \leq&C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}\Vert \Delta b\Vert _{L^{2}}^{1-\frac{s}{\beta}}\bigl\Vert \Lambda^{2+\beta}b\bigr\Vert _{L^{2}}^{1+\frac {s}{\beta}} \\ \leq&\frac{1}{8}\bigl\Vert \Lambda^{2+\beta}b\bigr\Vert _{L^{2}}^{2}+C\Vert \nabla b\Vert _{\dot{B}_{\infty,\infty}^{-\gamma_{2}}}^{\frac{2\beta}{2\beta-1-\gamma_{2}}}\Vert \Delta b\Vert _{L^{2}}^{2}. \end{aligned}$$

Inserting the above estimates into (2.6) and (2.7), and summing up the results and using the Gronwall inequality, we arrive at

$$\bigl\Vert (u,b)\bigr\Vert _{L^{\infty}(0,T;H^{2})}+\Vert u\Vert _{L^{2}(0,T;H^{2+\alpha})}+\Vert b\Vert _{L^{2}(0,T;H^{2+\beta})}\leq C. $$

This completes the proof.

Conclusions

The applications of Hall-MHD system cover a very wide range of physical objects, such as magnetic reconnection in space plasmas, star formation, neutron stars, and the geo-dynamo. In this paper, we obtained a new regularity criterion that improves and extends some known regularity criteria of the 3D generalized Hall-MHD system.

References

  1. Fan, J, Gao, H, Nakamura, G: Regularity criteria for the generalized magnetohydrodynamic equations and the quasi-geostrophic equations. Taiwan. J. Math. 15, 1059-1073 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Wu, J: Generalized MHD equations. J. Differ. Equ. 195, 284-312 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  3. Wu, J: Regularity criteria for the generalized MHD equatins. Commun. Partial Differ. Equ. 33, 285-306 (2008)

    Article  MATH  Google Scholar 

  4. Zhou, Y: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, 491-505 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  5. Fan, J, Malaikah, H, Monaquel, S, Nakamura, G, Zhou, Y: Global Cauchy problem of 2D generalized MHD equations. Monatshefte Math. 175(1), 127-131 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. Jiang, Z, Wang, Y, Zhou, Y: On regularity criteria for the 2D generalized MHD system. J. Math. Fluid Mech. 18(2), 331-341 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  7. Acheritogaray, M, Degond, P, Frouvelle, A, Liu, J: Kinetic formulation and global existence for the Hall-magnetohydrodynamics system. Kinet. Relat. Models 4, 901-918 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chae, D, Degond, P, Liu, J: Well-posedness for Hall-magnetohydrodynamics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31(3), 555-565 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  9. Chae, D, Schonbek, M: On the temporal decay for the Hall-magnetohydrodynamic equations. J. Differ. Equ. 255(11), 3971-3982 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. Wan, R, Zhou, Y: On the global existence, energy decay and blow up criterions for the Hall-MHD system. J. Differ. Equ. 259, 5982-6008 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  11. Zhang, Z: A remark on the blow-up criterion for the 3D Hall-MHD system in Besov spaces. Preprint (2015)

  12. Pan, N, Ma, C, Zhu, M: Global regularity for the 3D generalized Hall-MHD system. Appl. Math. Lett. 61, 62-66 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  13. Fan, J, Alsaedi, A, Hayat, T, Nakamura, G, Zhou, Y: On strong solutions to the compressible Hall-magnetohydrodynamic system. Nonlinear Anal., Real World Appl. 22, 423-434 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  14. Fan, J, Jia, X, Nakamura, G, Zhou, Y: On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z. Angew. Math. Phys. 66(4), 1695-1706 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  15. Fan, J, Ahmad, B, Hayat, T, Zhou, Y: On blow-up criteria for a new Hall-MHD system. Appl. Math. Comput. 274, 20-24 (2016)

    MathSciNet  Google Scholar 

  16. Fan, J, Ahmad, B, Hayat, T, Zhou, Y: On well-posedness and blow-up for the full compressible Hall-MHD system. Nonlinear Anal., Real World Appl. 31, 569-579 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  17. He, F, Ahmad, B, Hayat, T, Zhou, Y: On regularity criteria for the 3D Hall-MHD equations in terms of the velocity. Nonlinear Anal., Real World Appl. 32, 35-51 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  18. Wan, R, Zhou, Y: Low regularity well-posedness for the 3D generalized Hall-MHD system. Acta Appl. Math. (2016). doi:10.1007/s10440-016-0070-5

    Google Scholar 

  19. Jiang, Z, Zhu, M: Regularity criteria for the 3D generalized MHD and Hall-MHD systems. Bull. Malays. Math. Soc. (2015). doi:10.1007/s40840-015-0243-9

    Google Scholar 

  20. Ye, Z: Regularity criteria and small data global existence to the generalized viscous Hall-magnetohydrodynamics. Comput. Math. Appl. 70, 2137-2154 (2015)

    MathSciNet  Article  Google Scholar 

  21. Kato, T, Ponce, G: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891-907 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  22. Hajaiej, H, Molinet, L, Ozawa, T, Wang, B: Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations. RIMS Kôkyûroku Bessatsu 26, 159-175 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Machihara, S, Ozawa, T: Interpolation inequalities in Besov spaces. Proc. Am. Math. Soc. 131, 1553-1556 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  24. Meyer, Y: Oscillation patterns in some nonlinear evolution equations. In: Cannone, M, Miyakawa, T (eds.) Mathematical Foundation of Turbulent Viscous Flows. Lecture Notes in Mathematics, vol. 1871, pp. 101-187. Springer, Berlin (2006)

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the editors for their valuable suggestions and comments, which improved the results and the quality of the paper. Ma is partially supported by NSFC (Grant No. 11661070), the Youth Science and Technology Support Program (Grant No. 1606RJYE237) and the Scientific Research Foundation of the Higher Education Institutions (Grant No. 2015A-131) of Gansu Province.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Caochuan Ma.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gu, W., Ma, C. & Sun, J. A regularity criterion for the generalized Hall-MHD system. Bound Value Probl 2016, 188 (2016). https://doi.org/10.1186/s13661-016-0695-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-016-0695-3

MSC

  • 35Q35
  • 35B65
  • 76D05

Keywords

  • regularity criterion
  • generalized Hall-MHD
  • Gagliardo-Nirenberg inequality