Open Access

A regularity criterion for the generalized Hall-MHD system

Boundary Value Problems20162016:188

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

Received: 29 August 2016

Accepted: 13 October 2016

Published: 24 October 2016

Abstract

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

Keywords

regularity criterion generalized Hall-MHD Gagliardo-Nirenberg inequality

MSC

35Q35 35B65 76D05

1 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 \(q_{5}:=\frac{2}{1-\theta_{3}}\).

2 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.

3 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.

Declarations

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.

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.

Authors’ Affiliations

(1)
Center for Information, Nanjing Forestry University
(2)
Department of Mathematics, Tianshui Normal University
(3)
Department of Mathematics, Zhejiang Normal University
(4)
Department of Applied Mathematics, Nanjing Forestry University

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