A regularity criterion for the generalized Hall-MHD system

Abstract

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

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 [1–6].

When $\alpha=\beta=1$, the system (1.1)-(1.4) reduces to the well-known Hall-MHD system, which has received many studies [7–11]. 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 [10–12]. Some other relevant results about Hall-MHD equations can be found in [13–17].

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, ℓ, 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 [22–24]:

$$\begin{aligned}& \Vert \nabla u\Vert _{L^{3}}^{3}\leq C\Vert \nabla u \Vert _{\dot{B}_{\infty,\infty }^{-\gamma_{1}}}\Vert \nabla u\Vert _{\dot{H}^{\frac{\gamma_{1}}{2}}}^{2}, \end{aligned}$$

(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.