Serrin–type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

Hui Chen School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China chenhui@zust.edu.cn , Chenyin Qian Department of Mathematics, Zhejiang Normal University Jinhua, 321004, China qcyjcsx@163.com and Ting Zhang* School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China zhangting79@zju.edu.cn

Abstract.

In this paper, we consider the Cauchy problem to the 3D MHD equations. We show that the Serrin–type conditions imposed on one component of the velocity $u_{3}$ and one component of magnetic fields $b_{3}$ with

u_{3}\in L^{p_{0},1}(-1,0;L^{q_{0}}(B(2))),\ b_{3}\in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))),

$\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1$ and $3<q_{0},q_{1}<+\infty$ imply that the suitable weak solution is regular at $(0,0)$ . The proof is based on the new local energy estimates introduced by Chae-Wolf (Arch. Ration. Mech. Anal. 2021) and Wang-Wu-Zhang (arXiv:2005.11906).

^∗Corresponding author.

Key words: MHD equations; Serrin–type conditions; suitable weak solution; partial regularity; local energy estimates

AMS Subject Classification (2000): 35Q35; 35Q30; 76D03

1. Introduction

We consider the Cauchy problem to the three-dimensional incompressible magnetohydrodynamics (MHD) equations in $\mathbb{R}^{3}\times(0,\infty)$

(1.1)

\left\{\begin{array}[]{ll}\vspace{4pt}&\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}-\mu\Delta\bm{u}+\nabla\pi=\bm{b}\cdot\nabla\bm{b},\\ \vspace{4pt}&\partial_{t}\bm{b}+(\bm{u}\cdot\nabla)\bm{b}-\nu\Delta\bm{b}=\bm{b}\cdot\nabla\bm{u},\\ &\nabla\cdot\bm{u}=\nabla\cdot\bm{b}=0~{},\end{array}\right.

with initial data

(1.2)

\displaystyle\bm{u}|_{t=0}=\bm{u}_{0},~{}\bm{b}|_{t=0}=\bm{b}_{0}.

Here $\bm{u}=(u_{1},u_{2},u_{3}),\ \bm{b}=(b_{1},b_{2},b_{3})$ and $\pi$ are nondimensional quantities corresponding to the velocity, magnetic fields and a scalar pressure. $\mu>0$ is the viscosity coefficient and $\nu>0$ is the resistivity coefficient. We require $\mu=\nu=1$ in this paper.

Such system (1.1)–(1.2) describes many phenomena such as the geomagnetic dynamo in geophysics, solar winds and solar flares in astrophysics. G. Duvaut and J. L. Lions [8] constructed a global weak solution and the local strong solution to the initial boundary value problem, and the properties of such solutions have been examined by M. Sermange and R. Temam in [20].

If the magnetic fields $\bm{b}=0$ , the system (1.1) degenerates to the incompressible Navier–Stokes equations. A global weak solution to the Navier–Stokes equations was constructed by J. Leray [16]. However, the uniqueness and regularity of such weak solution is still one of the most challenging open problems in the field of mathematical fluid mechanics. One essential work is usually referred as Serrin–type conditions (see [9, 18, 21, 22] and the references therein.), i.e. if the weak solution $\bm{u}$ satisfies

(1.3)

\bm{u}\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \frac{2}{p}+\frac{3}{q}=1,\ 3\leqslant q\leqslant\infty,

then the weak solution is regular in $(0,T]$ . There are several notable results [3, 4, 5, 6, 11, 24] to weaken the above criteria by imposing constraints only on partial components or directional derivatives of velocity field. In particular, D. Chae and J. Wolf [3] made an important progress and obtained the regularity of solution under the condition

(1.4)

\displaystyle u_{3}\in L^{p}\left(0,T;L^{q}\left(\mathbb{R}^{3}\right)\right),\quad\frac{2}{p}+\frac{3}{q}<1,\quad 3<q\leqslant\infty.

W. Wang, D. Wu and Z. Zhang [24] improved to

(1.5)

\displaystyle u_{3}\in L^{p,1}\left(0,T;L^{q}\left(\mathbb{R}^{3}\right)\right),\quad\frac{2}{p}+\frac{3}{q}=1,\quad 3<q<\infty.

Throughout this paper, $L^{p,1}$ denotes the Lorentz space with respect to the time variable.

If the magnetic fields $\bm{b}\neq 0$ , the situation is more complicated due to the coupling effect between the velocity $\bm{u}$ and the magnetic fields $\bm{b}$ . Some fundamental Serrin–type regularity criteria in term of the velocity only were done in [7, 13, 26, 28]. For instance, if the weak solution $\left(\bm{u},\bm{b}\right)$ satisfies

(1.6)

\bm{u}\in L^{p}(0,T;L^{q}(\mathbb{R}^{3})),\ \ \frac{2}{p}+\frac{3}{q}=1,\ 3<q\leqslant\infty,

then the solution is regular in $(0,T]$ . More regularity criteria can be found in [12, 27] and the references therein.

We now recall the notion of suitable weak solution to the MHD equations (1.1).

Definition 1.1.

We say that $(\bm{u},\bm{b},\pi)$ is a suitable weak solution to the MHD equations (1.1) in an open domain $\Omega_{T}=\Omega\times(-T,0)$ , if

$(1)$

$\bm{u},\bm{b}\in L^{\infty}\left(-T,0;L^{2}(\Omega)\right)\cap L^{2}\left(-T,0;H^{1}(\Omega)\right)$ and $\pi\in L^{\frac{3}{2}}\left(\Omega_{T}\right)$ ;
$(2)$

(1.1) is satisfied in the sense of distribution;

(3)

the local energy inequality holds: for any nonnegative test function
$\varphi\in C_{c}^{\infty}\left(\Omega_{T}\right)$ and $t\in(-T,0)$ ,

		$\displaystyle\int_{\Omega}\left(\|\bm{u}\|^{2}+\|\bm{b}\|^{2}\right)\varphi\,{\rm d}x+2\int_{-T}^{t}\int_{\Omega}\left(\|\nabla\bm{u}\|^{2}+\|\nabla\bm{b}\|^{2}\right)\varphi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\leqslant$	$\displaystyle\int_{-T}^{t}\int_{\Omega}\left(\|\bm{u}\|^{2}+\|\bm{b}\|^{2}\right)\cdot\left(\partial_{s}\varphi+\Delta\varphi\right)+\bm{u}\cdot\nabla\varphi\left(\|\bm{u}\|^{2}+\|\bm{b}\|^{2}+2\pi\right)\,{\rm d}x\,{\rm d}s$
(1.7)			$\displaystyle-2\int_{-T}^{t}\int_{\Omega}\left(\bm{u}\cdot\bm{b}\right)\bm{b}\cdot\nabla\varphi\,{\rm d}x\,{\rm d}s.$

The global existence of suitable weak solution to the MHD equations (1.1) was investigated by C. He and Z. Xin [14]. They also obtained that one-dimensional Hausdorff measure of the possible space-time singular points set for the suitable weak solution is zero.

In this paper, inspired by [3, 24], we focus on the regularity criteria to the MHD equation (1.1) involving the terms $u_{3}$ and $b_{3}$ only.

Theorem 1.2.

Let $(\bm{u},\bm{b},\pi)$ be a suitable weak solution of the MHD equations (1.1) in $\mathbb{R}^{3}\times(-1,0).$ If $\bm{u},\bm{b}$ satisfies

(1.8)

u_{3}\in L^{p_{0},1}\left(-1,0;L^{q_{0}}(B(2))\right)

and

(1.9)

b_{3}\in L^{p_{1},1}\left(-1,0;L^{q_{1}}(B(2))\right)

with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1,~{}3<q_{0},q_{1}<+\infty$ , then the solution $(\bm{u},\bm{b})$ is regular at $(0,0)$ . Here $B(r)$ is the ball in $\mathbb{R}^{3}$ with center at origin and radius $r$ .

Remark 1.3.

It should be pointed out that the condition $\mu=\nu$ in the MHD equations (1.1) is essential in our estimates. Actually, we adopt the Elsässer variables $\bm{Z}^{+}=\bm{u}+\bm{b}$ and $\bm{Z}^{-}=\bm{u}-\bm{b}$ to obtain the crucial lemma 4.1 that if $u_{3}=b_{3}=0$ , the solution is regular at $(0,0)$ . However, it is not trivial in the case $\mu\neq\nu$ . We will discuss it in our future work.

Remark 1.4.

If we replace (1.8) with the subcritical regularity criteria

(1.10)

u_{3}\in L^{p_{0}}\left(-1,0;L^{q_{0}}(B(2))\right),\ \frac{2}{p_{0}}+\frac{3}{q_{0}}<1,\ 3<q_{0}\leqslant+\infty,

Theorem 1.2 still holds. Actually, we can pick $2<{p}_{2}<p_{0}$ and $3<q_{2}<q_{0}$ with $\frac{2}{p_{2}}+\frac{3}{q_{2}}=1$ . Therefore, we can prove it directly by the embedding inequality

\left\|u_{3}\right\|_{L^{p_{2},1}\left(-1,0;L^{q_{2}}(B(2))\right)}\leqslant C\left\|u_{3}\right\|_{L^{p_{0}}\left(-1,0;L^{q_{0}}(B(2))\right)}.

Analogously, we can replace (1.9) with the regularity criteria

(1.11)

b_{3}\in L^{p_{1}}\left(-1,0;L^{q_{1}}(B(2))\right),\ \frac{2}{p_{1}}+\frac{3}{q_{1}}<1,\ 3<q_{1}\leqslant+\infty.

Remark 1.5.

By the standard interpolation theory, it is possible to extend the regularity criteria in Theorem 1.2 to weaker one, such as

(1.12)

u_{3}\in L^{p_{0},1}\left(-1,0;L^{q_{0},\infty}(B(2))\right),\ b_{3}\in L^{p_{1},1}\left(-1,0;L^{q_{1},\infty}(B(2))\right),

with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1,~{}3<q_{0},q_{1}<+\infty$ . We leave the details to the interested readers.

Thanks to Theorem 1.2 and Remark 1.4, we obtain the following theorem.

Theorem 1.6.

Assume that $\bm{u}_{0},\ \bm{b}_{0}\in H^{1}(\mathbb{R}^{3})$ with $\textrm{div}\ \bm{u}_{0}=\textrm{div}\ \bm{b}_{0}=0$ and $(\bm{u},\bm{b})$ is the weak solution to the MHD equations (1.1)–(1.2). Then the solution is regular in $\mathbb{R}^{3}\times(0,T]$ , provided that

$\mathrm{(1)}$

$u_{3}\in L^{p_{0},1}(0,T;L^{q_{0}}(\mathbb{R}^{3}))$ with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=1$ , $3<q_{0}<\infty$ , or

$u_{3}\in L^{p_{0}}(0,T;L^{q_{0}}(\mathbb{R}^{3}))$ with $\frac{2}{p_{0}}+\frac{3}{q_{0}}<1$ , $3<q_{0}\leqslant+\infty$ ;
$\mathrm{(2)}$

$b_{3}\in L^{p_{1},1}(0,T;L^{q_{1}}(\mathbb{R}^{3}))$ with $\frac{2}{p_{1}}+\frac{3}{q_{1}}=1$ , $3<q_{1}<\infty$ , or

$b_{3}\in L^{p_{1}}(0,T;L^{q_{1}}(\mathbb{R}^{3}))$ with $\frac{2}{p_{1}}+\frac{3}{q_{1}}<1$ , $3<q_{1}\leqslant+\infty$ .

Our paper is organized as follows: we recall some notations and preliminary results in Section 2, and establish a key lemma in Section 3; finally, we will complete the proof of Theorem 1.2 in Section 4.

2. Notations and Preliminary

In this preparation section, we recall some usual notations and preliminary results.

For two comparable quantities, the inequality $X\lesssim Y$ stands for $X\leqslant CY$ for some positive constant $C$ . The dependence of the constant $C$ on other parameters or constants are usually clear from the context, and we will often suppress this dependence.

We use the following standard notations in the literature: Given matrix $\bm{A}=(A_{ij})\in\mathbb{R}^{m\times n}$ , we denote the norm $|\bm{A}|=\left(\Sigma_{i=1}^{m}\Sigma_{j=1}^{n}A_{ij}^{2}\right)^{\frac{1}{2}}$ ; for two vectors $\bm{a}=(a_{1},a_{2},a_{3}),\bm{b}=(b_{1},b_{2},b_{3})$ , $\bm{a}\cdot\bm{b}$ and $|\bm{a}|$ are the usual scalar product and norm in $\mathbb{R}^{3}$ respectively, and $\bm{a}\otimes\bm{b}$ is a matrix with $\left(\bm{a}\otimes\bm{b}\right)_{ij}=a_{i}b_{j}$ , $\left(\bm{a},\bm{b}\right)=(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$ ; for $x=(x_{1},x_{2},x_{3})$ $\in\mathbb{R}^{3}$ , $x^{\prime}=(x_{1},x_{2})$ is the horizontal variable; $B(x_{0},R)$ is the ball in $\mathbb{R}^{3}$ with center at $x_{0}$ and radius $R$ , and $B(R):=B(0,R)$ ; $B^{\prime}(x^{\prime}_{0},R)\subset\mathbb{R}^{2}$ is the ball in the horizontal plane with center at $x^{\prime}_{0}$ and radius $R$ , and $B^{\prime}(R):=B^{\prime}(0,R)$ ; $Q(z_{0},R)=B(x_{0},R)\times(t_{0}-R^{2},t_{0})$ with $z_{0}=(x_{0},t_{0})$ and $Q(R):=Q(0,R)$ ; we denote the integral mean value $\left(f(t)\right)_{B(r)}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\!\int_{B(r)}f(y,t)\,{\rm d}y$ .

We also shall use the same notation as that in Chae-Wolf [3]. Set

U_{n}(R)\buildrel\hbox{\footnotesize def}\over{=}B^{\prime}(R)\times(-r_{n},r_{n}),\ \ Q_{n}(R)\buildrel\hbox{\footnotesize def}\over{=}U_{n}(R)\times(-r_{n}^{2},0),

and

A_{n}(R)\buildrel\hbox{\footnotesize def}\over{=}Q_{n}(R)\backslash Q_{n+1}(R),

where $r_{n}=2^{-n},\ n\in\mathbb{N}$ . We consider $\mathrm{\Phi}_{n}$ given by

\mathrm{\Phi}_{n}(x,t)\buildrel\hbox{\footnotesize def}\over{=}\frac{1}{\sqrt{4\pi(-t+r_{n}^{2})}}e^{-\frac{x_{3}^{2}}{4(-t+r_{n}^{2})}},\quad(x,t)\in\mathbb{R}^{3}\times(-\infty,0),

which satisfies the fundamental solution of the backward heat equation

\partial_{t}\mathrm{\Phi}_{n}+\partial_{3}^{2}\mathrm{\Phi}_{n}=0.

There exist absolute constants $c_{1},c_{2}>0$ such that for $j=1,\ldots,n-1$ , it holds

(2.1)

\begin{split}&\mathrm{\Phi}_{n}\leqslant c_{2}r_{j}^{-1},\qquad\qquad\ \ |\partial_{3}\mathrm{\Phi}_{n}|\leqslant c_{2}r_{j}^{-2},\qquad\quad\text{in}\quad A_{j}(R),\\ c_{1}r_{n}^{-1}\leqslant&\mathrm{\Phi}_{n}\leqslant c_{2}r_{n}^{-1},\quad c_{1}r_{n}^{-2}\leqslant|\partial_{3}\mathrm{\Phi}_{n}|\leqslant c_{2}r_{n}^{-2},\quad\qquad\text{in}\quad Q_{n}(R).\end{split}

We denote the energy

	$\displaystyle E_{n}(R)$	$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}\sup_{t\in(-r_{n}^{2},0)}\int_{U_{n}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x+\int_{-r^{2}_{n}}^{0}\int_{U_{n}(R)}\|\nabla\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}t,$
	$\displaystyle\mathcal{E}$	$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}\sup_{t\in(-1,0)}\int_{\mathbb{R}^{3}}\|\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x+\int_{-1}^{0}\int_{\mathbb{R}^{3}}\|\nabla\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}t.$

The following lemma ensures the energy estimates.

Lemma 2.1.

[3, Lemma 3.1] Let $R\geqslant\frac{1}{2}$ . For $2\leqslant p\leqslant\infty$ , $2\leqslant q\leqslant 6$ , $\frac{2}{p}+\frac{3}{q}=\frac{3}{2}$ , we have

(2.2)

\|\bm{u}\|_{L^{p}\left(-r_{n}^{2},0;L^{q}\left(U_{n}(R)\right)\right)}^{2}+\|\bm{b}\|_{L^{p}\left(-r_{n}^{2},0;L^{q}\left(U_{n}(R)\right)\right)}^{2}\leqslant CE_{n}(R).

3. A Key Lemma

In this section, we apply a similar argument in Cafferalli-Kohn-Nirenberg [1], Chae-Wolf [3], or Wang-Wu-Zhang [24] to prove the following lemma.

Lemma 3.1.

Let $(\bm{u},\bm{b},\pi)$ be a suitable weak solution of (1.1) in $\mathbb{R}^{3}\times(-1,0).$ If $\bm{u},\bm{b}$ satisfies

(3.1)

u_{3}\in L^{p_{0},1}\left(-1,0;L^{q_{0}}(B(2))\right),\ b_{3}\in L^{p_{1},1}\left(-1,0;L^{q_{1}}(B(2))\right),

with $\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1,~{}3<q_{0},q_{1}<+\infty$ , then

\displaystyle r^{-2}\|\bm{u}\|_{L^{3}\left(Q(r)\right)}^{3}+r^{-2}\|\bm{b}\|_{L^{3}\left(Q(r)\right)}^{3}\leqslant C,

for any $0<r\leqslant 1$ .

For fixed $p_{0},p_{1}$ , we can pick $\frac{4q_{0}}{2q_{0}-5}<p_{0}^{*}<p_{0}$ and $2<p_{1}^{*}<p_{1}$ . Set

(3.2)

\displaystyle\mathscr{B}_{i}=r_{i}^{1-\frac{2}{p_{0}^{*}}-\frac{3}{q_{0}}}\|u_{3}\|_{L^{p_{0}^{*}}\left(-r_{i}^{2},0;L^{q_{0}}(B(2))\right)}+r_{i}^{1-\frac{2}{p_{1}^{*}}-\frac{3}{q_{1}}}\|b_{3}\|_{L^{p_{1}^{*}}\left(-r_{i}^{2},0;L^{q_{1}}(B(2))\right)}.

By Lemma A.5, we have

(3.3)

\displaystyle\sum_{i=0}^{+\infty}\mathscr{B}_{i}\leqslant C\left(\left\|u_{3}\right\|_{L^{p_{0},1}\left(-1,0;L^{q_{0}}(B(2))\right)}+\left\|b_{3}\right\|_{L^{p_{1},1}\left(-1,0;L^{q_{1}}(B(2))\right)}\right).

Let $\eta\left(x_{3},t\right)\in C_{c}^{\infty}((-1,1)\times(-1,0])$ denotes a cut-off function, $0\leqslant\eta\leqslant 1$ , and $\eta=1$ on $\left(-\frac{1}{2},\frac{1}{2}\right)\times\left(-\frac{1}{4},0\right)$ .

In addition, let $\frac{1}{2}\leqslant\rho<R\leqslant 1$ be arbitrarily chosen, but $|R-\rho|\leqslant\frac{1}{2}.$ Let $\psi=\psi\left(x^{\prime}\right)\in C^{\infty}\left(\mathbb{R}^{2}\right)$ with $0\leqslant\psi\leqslant 1$ in $B^{\prime}(R)$ satisfying

(3.4)

\psi(x^{\prime})=\psi(|x^{\prime}|)=\left\{\begin{array}[]{l}1\text{ in }B^{\prime}(\rho)\\ 0\text{ in }\mathbb{R}^{2}\backslash B^{\prime}\left(\frac{R+\rho}{2}\right)\end{array}\right.,

and

|D\psi|\leqslant\frac{C}{R-\rho},\quad\left|D^{2}\psi\right|\leqslant\frac{C}{(R-\rho)^{2}}.

For $j=0,1,\cdots,n$ , denote $\chi_{j}=\chi_{B^{\prime}(R)}(x^{\prime})\cdot\eta(2^{j}\cdot x_{3},2^{2j}\cdot t)$ , where $\chi_{B^{\prime}(R)}$ is the indicator function of the set $B^{\prime}(R)$ . Let

(3.5)

\begin{array}[]{lll}\phi_{j}=\left\{\begin{array}[]{ll}\chi_{j}-\chi_{j+1},&\text{ if }\quad j=0,\ldots,n-1,\\ \chi_{n},&\text{ if }\quad j=n.\end{array}\right.\end{array}

Taking the test function $\varphi=\mathrm{\Phi}_{n}\eta\psi$ in ( $(3)$ ), we have

		$\displaystyle\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \mathrm{\Phi}_{n}\eta\psi\,{\rm d}x+2\int_{-1}^{t}\int_{U_{0}(R)}\|\nabla\left(\bm{u},\bm{b}\right)\|^{2}\ \mathrm{\Phi}_{n}\eta\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\leqslant$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\cdot\left(\partial_{s}+\Delta\right)\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
		$\displaystyle+\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)-2\left(\bm{u}\cdot\bm{b}\right)\bm{b}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
(3.6)			$\displaystyle+2\int_{-1}^{t}\int_{U_{0}(R)}\pi\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s.$

Next, we shall handle the right side of (3) term by term.

3.1. Estimates for nonlinear terms

Lemma 3.2.

Under the assumptions of Lemma 3.1, we have

(3.7)

\int_{-1}^{t}\int_{U_{0}(R)}|\left(\bm{u},\bm{b}\right)|^{2}\cdot\left(\partial_{s}+\Delta\right)\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s\leqslant C\frac{\mathcal{E}}{(R-\rho)^{2}}.

Proof.

By the estimates (2.1) for $\mathrm{\Phi}_{n}$ and the fact that $supp\ \nabla\eta\subseteq A_{0},supp\ \phi_{j}\subseteq Q_{j}\backslash Q_{j+2}$ , we have

		$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\cdot\left(\partial_{s}+\Delta\right)\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\cdot\left(\mathrm{\Phi}_{n}\partial_{s}\eta\psi+2\partial_{3}\mathrm{\Phi}_{n}\partial_{3}\eta\psi+\mathrm{\Phi}_{n}\Delta(\eta\psi)\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\int_{-1}^{0}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}s+(R-\rho)^{-2}\sum_{i=0}^{n}\int_{-1}^{0}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \mathrm{\Phi}_{n}\phi_{i}\,{\rm d}x\,{\rm d}s$
	$\displaystyle\leqslant$	$\displaystyle\mathcal{E}+(R-\rho)^{-2}\sum_{i=0}^{n}r_{i}^{-1}\int_{-r_{i}^{2}}^{0}\int_{U_{i}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\mathcal{E}+(R-\rho)^{-2}\sum_{i=0}^{n}r_{i}\mathcal{E}$
	$\displaystyle\lesssim$	$\displaystyle(R-\rho)^{-2}\mathcal{E}.$

∎∎

Lemma 3.3.

Under the assumptions of Lemma 3.1, we have

		$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)-2\left(\bm{u}\cdot\bm{b}\right)\bm{b}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
(3.8)		$\displaystyle\leqslant$	$\displaystyle C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\mathcal{E}^{\frac{3}{2}}+C(R-\rho)^{-1}\ \mathcal{E}^{\frac{1}{2}}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right),$

where $\mathscr{B}_{i}$ is defined in (3.2).

Proof.

We first note that

		$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)-2\left(\bm{u}\cdot\bm{b}\right)\bm{b}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\cdot\left(\eta\psi\right)\,{\rm d}x\,{\rm d}s$
		$\displaystyle+\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\mathrm{\Phi}_{n}\nabla\left(\eta\psi\right)\,{\rm d}x\,{\rm d}s$
		$\displaystyle-2\int_{-1}^{t}\int_{U_{0}(R)}\left(\bm{u}\cdot\bm{b}\right)b_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\cdot\left(\eta\psi\right)\,{\rm d}x\,{\rm d}s$
		$\displaystyle-2\int_{-1}^{t}\int_{U_{0}(R)}\left(\bm{u}\cdot\bm{b}\right)\bm{b}\cdot\mathrm{\Phi}_{n}\nabla\left(\eta\psi\right)\,{\rm d}x\,{\rm d}s$
(3.9)		$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}$	$\displaystyle I_{1}+I_{2}+I_{3}+I_{4}.$

By the estimates (2.1) for $\mathrm{\Phi}_{n}$ , Hölder inequality and Lemma 2.1, we have

	$\displaystyle I_{1}=$	$\displaystyle\sum_{i=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\cdot\left(\phi_{i}\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}r_{i}^{-2}\int_{-r_{i}^{2}}^{0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{2q_{0}}{q_{0}-1}}(U_{i}(R))}^{2}\\|u_{3}\\|_{L^{q_{0}}(U_{i}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}r_{i}^{-\frac{2}{p_{0}^{}}-\frac{3}{q_{0}}}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{4q_{0}}{3}}\left(-r_{i}^{2},0;L^{\frac{2q_{0}}{q_{0}-1}}(U_{i}(R))\right)}^{2}\\|u_{3}\\|_{L^{p_{0}^{}}\left(-r_{i}^{2},0;L^{q_{0}}(U_{i}(R))\right)}$
(3.10)		$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right).$

For the second term, by (2.1) and Lemma 2.1 again, we have

	$\displaystyle I_{2}=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\mathrm{\Phi}_{n}\nabla\eta\psi+\|\left(\bm{u},\bm{b}\right)\|^{2}\ \bm{u}\cdot\mathrm{\Phi}_{n}\eta\nabla\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\int_{-1}^{0}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{3}\,{\rm d}x\,{\rm d}s+\frac{1}{(R-\rho)}\sum_{i=0}^{n}r_{i}^{-1}\int_{-r_{i}^{2}}^{0}\int_{U_{i}(R)}\|\left(\bm{u},\bm{b}\right)\|^{3}\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\mathcal{E}^{\frac{3}{2}}+\frac{1}{(R-\rho)}\sum_{i=0}^{n}r_{i}^{-1}r_{i}^{\frac{1}{2}}\ \\|\left(\bm{u},\bm{b}\right)\\|_{L^{4}\left(-r_{i}^{2},0;L^{3}\left(U_{i}(R)\right)\right)}^{3}$
(3.11)		$\displaystyle\lesssim$	$\displaystyle\mathcal{E}^{\frac{3}{2}}+\frac{\mathcal{E}^{\frac{1}{2}}}{(R-\rho)}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right).$

As to $I_{3},I_{4}$ , we have

	$\displaystyle I_{3}\lesssim$	$\displaystyle\sum_{i=0}^{n}r_{i}^{-2}\int_{-r_{i}^{2}}^{0}\ \\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{2q_{1}}{q_{1}-1}}(U_{i}(R))}^{2}\\|b_{3}\\|_{L^{q_{1}}(U_{i}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}r_{i}^{-\frac{2}{p_{1}^{}}-\frac{3}{q_{1}}}\ \\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{4q_{1}}{3}}\left(-r_{i}^{2},0;L^{\frac{2q_{1}}{q_{1}-1}}(U_{i}(R))\right)}^{2}\\|b_{3}\\|_{L^{p_{1}^{}}\left(-r_{i}^{2},0;L^{q_{1}}(U_{i}(R))\right)}$
(3.12)		$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right),$

which is analogous to (3.1), and

	$\displaystyle I_{4}\lesssim$	$\displaystyle\int_{-1}^{0}\int_{U_{0}(R)}\|\left(\bm{u},\bm{b}\right)\|^{3}\,{\rm d}x\,{\rm d}s+\frac{1}{(R-\rho)}\sum_{i=0}^{n}r_{i}^{-1}\int_{-r_{i}^{2}}^{0}\int_{U_{i}(R)}\|\left(\bm{u},\bm{b}\right)\|^{3}\,{\rm d}x\,{\rm d}s$
(3.13)		$\displaystyle\lesssim$	$\displaystyle\mathcal{E}^{\frac{3}{2}}+\frac{\mathcal{E}^{\frac{1}{2}}}{(R-\rho)}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right),$

which is analogous to (3.1). Combining (3.1) with (3.1), (3.1), (3.1) and (3.1), we obtain (3.3). ∎∎

3.2. Estimates for the pressure

This part is devoted to the estimates regarding the pressure term on the right side of (3). We will give a similar argument in [24] here.

Given $3\times 3$ matrix valued function $f=\left(f_{jk}\right)$ , we set

\displaystyle\widehat{\mathscr{J}(f)}(\xi)=\sum_{j,k=1,2,3}\frac{\xi_{j}\xi_{k}}{|\xi|^{2}}\mathcal{F}f_{jk},

where the Fourier transform is defined by

\mathcal{F}g(\xi)=\hat{g}(\xi)=\int_{\mathbb{R}^{3}}e^{-i(x\cdot\xi)}g(x)\,{\rm d}x.

Therefore, $\mathscr{J}:L^{q}\left(\mathbb{R}^{3}\right)\rightarrow L^{q}\left(\mathbb{R}^{3}\right)$ , $1<q<+\infty$ defines a bounded linear operator with

(3.14)

\displaystyle\|\mathscr{J}(f)\|_{L^{q}\left(\mathbb{R}^{3}\right)}\leqslant C\ \|f\|_{L^{q}\left(\mathbb{R}^{3}\right)}.

Denote $f=(f_{jk})=\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\chi_{0}$ ,

(3.15)

\displaystyle\pi_{0}=\mathscr{J}\left(f\right),\ \pi_{h}=\pi-\pi_{0}.

It follows that

\displaystyle\Delta\pi_{0}=\nabla\cdot\nabla\cdot f=\mathrm{\Sigma}_{j,k=1}^{3}\ \partial_{j}\partial_{k}f_{jk}\quad\text{ in }\quad\mathbb{R}^{3}\times(-1,0)

in the sense of distributions and $\pi_{h}$ is harmonic in $Q_{1}(R)$ . Then we have

		$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\pi\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s+\int_{-1}^{t}\int_{U_{0}(R)}\pi_{h}\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\eta\psi\,{\rm d}x\,{\rm d}s+\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\mathrm{\Phi}_{n}\nabla\left(\eta\psi\right)\,{\rm d}x\,{\rm d}s$
		$\displaystyle-\int_{-1}^{t}\int_{U_{0}(R)}\nabla\pi_{h}\cdot\bm{u}\cdot\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x\,{\rm d}s$
(3.16)		$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}$	$\displaystyle J+K+H.$

Lemma 3.4.

Under the assumptions of Lemma 3.1, we have

		$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\pi\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\eta\psi\right)\,{\rm d}x{\,{\rm d}s}$
(3.17)		$\displaystyle\leqslant$	$\displaystyle C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\frac{\mathcal{E}^{\frac{1}{2}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right)+\frac{C}{(R-\rho)^{3}}\mathcal{E}^{\frac{3}{2}}.$

Proof.

Combining the estimates (3.2), (A.2), (A.7) and (A.12), we have (3.4).∎∎

3.3. The proof of Lemma 3.1

On the basis of the estimates of the nonlinear term and the pressure in subsection 3.1–3.2, we are in position to give the detail proof of Lemma 3.1.

Proof.

Gathering (3) and the estimates in Lemma 3.2, 3.3 and 3.4, we have

	$\displaystyle r_{n}^{-1}E_{n}(\rho)\leqslant$	$\displaystyle C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\frac{\mathcal{E}^{\frac{1}{2}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right)+C\frac{1+\mathcal{E}^{\frac{3}{2}}}{(R-\rho)^{3}}$
	$\displaystyle\leqslant$	$\displaystyle C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\frac{\mathcal{E}^{\frac{3}{4}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{-\frac{5}{8}}E_{i}(R)^{\frac{3}{4}}r_{i}^{\frac{1}{8}}+C\frac{1+\mathcal{E}^{\frac{3}{2}}}{(R-\rho)^{3}}$
	$\displaystyle\leqslant$	$\displaystyle C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\frac{\mathcal{E}^{\frac{3}{4}}}{R-\rho}\left(\sum_{i=0}^{n}r_{i}^{-\frac{5}{6}}E_{i}(R)\right)^{\frac{3}{4}}\left(\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\right)^{\frac{1}{4}}$
		$\displaystyle+C\frac{1+\mathcal{E}^{\frac{3}{2}}}{(R-\rho)^{3}}$
	$\displaystyle\leqslant$	$\displaystyle C_{0}\sum_{i=0}^{n}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(R)\right)+C_{0}\frac{1+\mathcal{E}^{3}}{(R-\rho)^{4}}.$

In view of (3.3), there exists a sufficient large integer $n_{0}\geqslant 1$ such that

(3.18)

C_{0}\sum_{i=n_{0}}^{\infty}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\leqslant\frac{1}{2}.

Then for $n\geqslant n_{0}$ we have

	$\displaystyle r_{n}^{-1}E_{n}(\rho)\leqslant$	$\displaystyle C_{0}\sum_{i=n_{0}}^{n}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(R)\right)$
		$\displaystyle\quad+C_{0}\sum_{i=0}^{n_{0}-1}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(R)\right)+C_{0}\frac{1+\mathcal{E}^{3}}{(R-\rho)^{4}}$
(3.19)		$\displaystyle\leqslant$	$\displaystyle C_{0}\sum_{i=n_{0}}^{n}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(R)\right)+\frac{A_{0}}{(R-\rho)^{4}},$

where the constant

A_{0}=C_{1}\cdot 2^{n_{0}}\left(1+\left\|u_{3}\right\|_{L^{p_{0},1}\left(-1,0;L^{q_{0}}(B(2))\right)}^{2}+\left\|b_{3}\right\|_{L^{p_{1},1}\left(-1,0;L^{q_{1}}(B(2))\right)}^{2}+\mathcal{E}^{3}\right).

As the iteration argument in [10, V. Lemma 3.1], we introduce the sequence $\{\rho_{k}\}_{k=0}^{+\infty}$ which satisfy

\rho_{0}=\frac{1}{2},\quad\rho_{k+1}-\rho_{k}=\frac{1-\theta}{2}\theta^{k},

with $\frac{1}{2}<\theta^{4}<1$ and $\lim_{k\rightarrow\infty}\rho_{k}=1$ . For $n_{0}\leqslant j\leqslant n$ and $k\geqslant 1$ , we have

	$\displaystyle r_{j}^{-1}E_{j}(\rho_{k})\leqslant$	$\displaystyle C_{0}\sum_{i=n_{0}}^{j}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(\rho_{k+1})\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}\theta^{-4k}$
(3.20)		$\displaystyle\leqslant$	$\displaystyle C_{0}\sum_{i=n_{0}}^{n}\left(\mathscr{B}_{i}+r_{i}^{\frac{1}{6}}\right)\left(r_{i}^{-1}E_{i}(\rho_{k+1})\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}\theta^{-4k}.$

Applying (3.18) and (3.3) to the iteration argument from (3.3), we obtain that for $n\geqslant n_{0}$ ,

	$\displaystyle r_{n}^{-1}E_{n}(\rho_{0})\leqslant$	$\displaystyle C_{0}\sum_{j=n_{0}}^{n}\left(\mathscr{B}_{j}+r_{j}^{\frac{1}{6}}\right)\left(r_{j}^{-1}E_{j}(\rho_{1})\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}$
	$\displaystyle\leqslant$	$\displaystyle\frac{1}{2}C_{0}\sum_{j=n_{0}}^{n}\left(\mathscr{B}_{j}+r_{j}^{\frac{1}{6}}\right)\left(r_{j}^{-1}E_{j}(\rho_{2})\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}\left(1+\frac{1}{2\theta^{4}}\right)$
	$\displaystyle\leqslant$	$\displaystyle\frac{1}{2^{k-1}}C_{0}\sum_{j=n_{0}}^{n}\left(\mathscr{B}_{j}+r_{j}^{\frac{1}{6}}\right)\left(r_{j}^{-1}E_{j}(\rho_{k})\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}\sum_{j=0}^{k-1}\left(\frac{1}{2\theta^{4}}\right)^{j}$
	$\displaystyle\leqslant$	$\displaystyle\frac{1}{2^{k-1}}C_{0}\sum_{j=n_{0}}^{n}\left(\mathscr{B}_{j}+r_{j}^{\frac{1}{6}}\right)\left(r_{j}^{-1}E_{j}(1)\right)+A_{0}\cdot\frac{16}{(1-\theta)^{4}}\cdot\frac{2\theta^{4}}{2\theta^{4}-1}.$

Let $k\rightarrow\infty$ , we obtain that for $n\geqslant n_{0}$ ,

(3.21)

r_{n}^{-1}E_{n}(\rho_{0})\leqslant A_{0}\cdot\frac{16}{(1-\theta)^{4}}\cdot\frac{2\theta^{4}}{2\theta^{4}-1}.

For $0<r\leqslant r_{n_{0}}$ , there exists an integer $j\geqslant n_{0}$ , such that

r_{j+1}<r\leqslant r_{j},

which together with (3.21) ensures that

\displaystyle r^{-1}\sup_{t\in(-r^{2},0)}\int_{B(r)}|\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x+r^{-1}\int_{Q(r)}|\nabla\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x\,{\rm d}t\leqslant 2\ r_{j}^{-1}E_{j}(\rho_{0})\leqslant C.

For $r_{n_{0}}<r\leqslant 1$ ,

\displaystyle r^{-1}\sup_{t\in(-r^{2},0)}\int_{B(r)}|\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x+r^{-1}\int_{Q(r)}|\nabla\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x\,{\rm d}t\leqslant 2^{n_{0}}\mathcal{E}\leqslant C.

Hence for $0<r\leqslant 1$ ,

(3.22)

\displaystyle r^{-1}\sup_{t\in(-r^{2},0)}\int_{B(r)}|\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x+r^{-1}\int_{Q(r)}|\nabla\left(\bm{u},\bm{b}\right)|^{2}\,{\rm d}x\,{\rm d}t\leqslant C.

By Sobolev embedding theorem, we have that for $0<r\leqslant 1$ ,

(3.23)

\displaystyle r^{-2}\|\left(\bm{u},\bm{b}\right)\|_{L^{3}(Q(r))}^{3}\leqslant Cr^{-\frac{3}{2}}\|\left(\bm{u},\bm{b}\right)\|_{L^{4}\left(-r^{2},0;L^{3}(B(r))\right)}^{3}\leqslant C.

The proof is completed.∎∎

4. Proof of Theorem 1.2

Before starting the proof, we give an essential lemma for the 3D MHD equation (1.1).

Lemma 4.1.

Let $\mu=\nu=1$ in the MHD equations (1.1). Assume that $\left(\bm{u},\bm{b},\pi\right)$ is a suitable weak solution in $Q(1)$ with $u_{3}=b_{3}=0$ . Then the solution is regular in $\overline{Q(\frac{1}{2})}$ .

Proof.

Denote $\mathcal{S}_{\mathrm{MHD}}$ the singular set of the suitable weak solution, the cylinder $U(x,r)=B^{\prime}(x^{\prime},r)\times(x_{3}-r,x_{3}+r)$ and $U(r):=U(0,r)$ . By [14] or [23, Theorem 1.4], we know that the one-dimensional Hausdorff measure of $\mathcal{S}_{\mathrm{MHD}}$ is zero.

Assume that the assertion does not hold, i.e. $\mathcal{S}_{\mathrm{MHD}}\cap\overline{Q(\frac{1}{2})}\neq\varnothing$ . By a similar argument in [15, Lemma 3.2], there exists $(x_{0},t_{0})\in\overline{Q(\frac{1}{2})}$ and some $\varepsilon,\rho$ with $0<\varepsilon<\rho<\frac{1}{2}$ , such that

\displaystyle(x_{0},t_{0})\in\mathcal{S}_{\mathrm{MHD}}\cap\overline{U(x_{0},\rho)\times(t_{0}-\rho^{2},t_{0})}\subseteq U\left(x_{0},\rho-\varepsilon\right)\times\left\{t_{0}\right\}.

Without loss of generality, we may assume $(x_{0},t_{0})=(0,0)$ . Thus, the solution $(\bm{u},\bm{b},\nabla\pi)\in C^{\infty}(\overline{U(\rho)}\times[-\rho^{2},0))$ and is singular at $(0,0)$ . Moreover, there exists a constant $C$ , such that

(4.1)

\displaystyle\sup_{-1<t<0}\|\left(\bm{u},\bm{b}\right)\|_{L^{2}(B(1))}^{2}+\|\nabla\left(\bm{u},\bm{b}\right)\|_{L^{2}(Q(1))}^{2}+\|\pi\|_{L^{\frac{3}{2}}(Q(1))}\leqslant C,

and for $(x,t)\in U(\rho)\backslash U(\rho-\varepsilon)\times[-\rho^{2},0)$ or $(x,t)\in U(\rho)\times\{-\rho^{2}\}$ ,

(4.2)

\displaystyle|\bm{u}|+|\bm{b}|\leqslant C.

We introduce the Elsässer variables $\bm{Z}^{+}=\left(Z_{1}^{+},Z_{2}^{+},0\right)=\bm{u}+\bm{b}$ and $\bm{Z}^{-}=\left(Z_{1}^{-},Z_{2}^{-},0\right)=\bm{u}-\bm{b}$ , which satisfy

	$\displaystyle\partial_{t}\bm{Z}^{+}+\bm{Z}^{-}\cdot\nabla\bm{Z}^{+}-\Delta\bm{Z}^{+}+\nabla\pi=0,$
	$\displaystyle\partial_{t}\bm{Z}^{-}+\bm{Z}^{+}\cdot\nabla\bm{Z}^{-}-\Delta\bm{Z}^{-}+\nabla\pi=0.$

As in [2], we introduce $\overline{\bm{Z}^{+}}=\int_{-\rho}^{\rho}\bm{Z}^{+}\cdot\beta(x_{3})\,{\rm d}x_{3},\ \overline{\bm{Z}^{-}}=\int_{-\rho}^{\rho}\bm{Z}^{-}\cdot\beta(x_{3})\,{\rm d}x_{3}$ , where the non-negative cut-off function $\beta(x_{3})\in C_{c}^{\infty}((-\rho,\rho))$ , $\beta=\frac{1}{2\rho-\varepsilon}$ on $(-\rho+\varepsilon,\rho-\varepsilon)$ and $\int_{-\rho}^{\rho}\beta(x_{3})\,{\rm d}x_{3}=1$ . It is easy to find that $\partial_{3}\pi=0$ , and

	$\displaystyle\partial_{t}\overline{\bm{Z}^{+}}+\int_{-\rho}^{\rho}\bm{Z}^{-}\cdot\nabla\bm{Z}^{+}\cdot\beta\,{\rm d}x_{3}-\Delta\overline{\bm{Z}^{+}}+\nabla\pi$	$\displaystyle=\int_{-\rho}^{\rho}\bm{Z}^{+}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3},$
	$\displaystyle\partial_{t}\overline{\bm{Z}^{-}}+\int_{-\rho}^{\rho}\bm{Z}^{+}\cdot\nabla\bm{Z}^{-}\cdot\beta\,{\rm d}x_{3}-\Delta\overline{\bm{Z}^{-}}+\nabla\pi$	$\displaystyle=\int_{-\rho}^{\rho}\bm{Z}^{-}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}.$

Define $\widetilde{\bm{Z}^{+}}=\bm{Z}^{+}-\overline{\bm{Z}^{+}}$ , $\widetilde{\bm{Z}^{-}}=\bm{Z}^{-}-\overline{\bm{Z}^{-}}$ , which satisfy

(4.3)		$\displaystyle\partial_{t}\widetilde{\bm{Z}^{+}}+\bm{Z}^{-}\cdot\nabla\widetilde{\bm{Z}^{+}}+\widetilde{\bm{Z}^{-}}\cdot\nabla\overline{\bm{Z}^{+}}-\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{-}}\cdot\nabla\widetilde{\bm{Z}^{+}}\cdot\beta\,{\rm d}x_{3}-\Delta\widetilde{\bm{Z}^{+}}=$	$\displaystyle-\int_{-\rho}^{\rho}\bm{Z}^{+}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}$
(4.4)		$\displaystyle\partial_{t}\widetilde{\bm{Z}^{-}}+\bm{Z}^{+}\cdot\nabla\widetilde{\bm{Z}^{-}}+\widetilde{\bm{Z}^{+}}\cdot\nabla\overline{\bm{Z}^{-}}-\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{+}}\cdot\nabla\widetilde{\bm{Z}^{-}}\cdot\beta\,{\rm d}x_{3}-\Delta\widetilde{\bm{Z}^{-}}=$	$\displaystyle-\int_{-\rho}^{\rho}\bm{Z}^{-}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}.$

Let $\alpha(x_{1},x_{2})\in C_{c}^{\infty}(B^{\prime}(\rho))$ be a cut-off function with $0\leqslant\alpha\leqslant 1$ , $\alpha=1$ on $B^{\prime}(\rho-\varepsilon)$ . Multiplying (4.3) by $4|\widetilde{\bm{Z}^{+}}|^{2}\widetilde{\bm{Z}^{+}}\cdot\varphi^{4}$ with $\varphi(x)=\alpha(x_{1},x_{2})\cdot\beta(x_{3})$ , and integrating the resulting equations over $\mathbb{R}^{3}$ , applying (4.1), (4.2), Hölder’s inequality and Sobolev embedding theorem, we have

		$\displaystyle\frac{d}{dt}\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+4\int_{\mathbb{R}^{3}}\|\nabla\widetilde{\bm{Z}^{+}}\|^{2}\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\|^{2}\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}^{3}}\bm{Z}^{-}\cdot\nabla\alpha^{4}\cdot\beta^{4}\|\widetilde{\bm{Z}^{+}}\|^{4}-4\widetilde{\bm{Z}^{-}}\cdot\nabla\overline{\bm{Z}^{+}}\cdot\|\widetilde{\bm{Z}^{+}}\|^{2}\widetilde{\bm{Z}^{+}}\varphi^{4}\,{\rm d}x$
		$\displaystyle+4\int_{\mathbb{R}^{3}}\left(\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{-}}\cdot\nabla\widetilde{\bm{Z}^{+}}\cdot\beta-\bm{Z}^{+}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}\right)\cdot\|\widetilde{\bm{Z}^{+}}\|^{2}\widetilde{\bm{Z}^{+}}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\widetilde{\bm{Z}^{+}}\|^{4}\|\nabla\varphi^{2}\|^{2}\,{\rm d}x$
	$\displaystyle\lesssim$	$\displaystyle 1+\left(\\|\nabla\overline{\bm{Z}^{+}}\\|_{L^{2}(B^{\prime}(\rho))}+\\|\nabla\widetilde{\bm{Z}^{+}}\\|_{L^{2}(U(\rho))}\right)\cdot\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}$
		$\displaystyle\times\\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}+\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{3}$
	$\displaystyle\leqslant$	$\displaystyle C+C\left(1+\\|\nabla\bm{Z}^{+}\\|_{L^{2}(U(\rho))}^{2}\right)\cdot\left(\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}\right)$
(4.5)			$\displaystyle-\\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}^{2}.$

Analogously, multiplying (4.4) by $4|\widetilde{\bm{Z}^{-}}|^{2}\widetilde{\bm{Z}^{-}}\cdot\varphi^{4}$ , and integrating the resulting equations over $\mathbb{R}^{3}$ , we have

		$\displaystyle\frac{d}{dt}\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}+4\int_{\mathbb{R}^{3}}\|\nabla\widetilde{\bm{Z}^{-}}\|^{2}\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\nabla\left(\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{2}\right)\|^{2}\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}^{3}}\bm{Z}^{+}\cdot\nabla\alpha^{4}\cdot\beta^{4}\|\widetilde{\bm{Z}^{-}}\|^{4}-4\widetilde{\bm{Z}^{+}}\cdot\nabla\overline{\bm{Z}^{-}}\cdot\|\widetilde{\bm{Z}^{-}}\|^{2}\widetilde{\bm{Z}^{-}}\varphi^{4}\,{\rm d}x$
		$\displaystyle+4\int_{\mathbb{R}^{3}}\left(\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{+}}\cdot\nabla\widetilde{\bm{Z}^{-}}\cdot\beta-\bm{Z}^{-}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}\right)\cdot\|\widetilde{\bm{Z}^{-}}\|^{2}\widetilde{\bm{Z}^{-}}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\widetilde{\bm{Z}^{-}}\|^{4}\|\nabla\varphi^{2}\|^{2}\,{\rm d}x$
	$\displaystyle\leqslant$	$\displaystyle C+C\left(1+\\|\nabla\bm{Z}^{-}\\|_{L^{2}(U(\rho))}^{2}\right)\cdot\left(\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}\right)$
(4.6)			$\displaystyle-\\|\nabla\left(\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}^{2}.$

Summing up (4) and (4) and applying Gronwall’s inequality, we have that for all $-\rho^{2}<t<0$ ,

(4.7)

\displaystyle\|\widetilde{\bm{Z}^{+}}\|_{L^{4}(U(\rho-\varepsilon))}^{4}+\|\widetilde{\bm{Z}^{-}}\|_{L^{4}(U(\rho-\varepsilon))}^{4}\leqslant C\|\widetilde{\bm{Z}^{+}}\varphi\|_{L^{4}(U(\rho))}^{4}+C\|\widetilde{\bm{Z}^{-}}\varphi\|_{L^{4}(U(\rho))}^{4}\leqslant C.

Accordingly, by (4.2) and (4.7), we have that for $0<r<\rho-\varepsilon$ ,

		$\displaystyle r^{-2}\left(\\|\bm{u}\\|_{L^{3}(Q(r))}^{3}+\\|\bm{b}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{-2}\left(\\|\bm{Z}^{+}\\|_{L^{3}(Q(r))}^{3}+\\|\bm{Z}^{-}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{-2}\left(\\|\widetilde{\bm{Z}^{+}}\\|_{L^{3}(Q(r))}^{3}+\\|\widetilde{\bm{Z}^{-}}\\|_{L^{3}(Q(r))}^{3}\right)+Cr^{-2}\left(\\|\overline{\bm{Z}^{+}}\\|_{L^{3}(Q(r))}^{3}+\\|\overline{\bm{Z}^{-}}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{\frac{3}{4}}+Cr^{-1}\int_{-r^{2}}^{0}\int_{B^{\prime}(r)}\left(\|\overline{\bm{Z}^{+}}\|^{3}+\|\overline{\bm{Z}^{-}}\|^{3}\right)\,{\rm d}x_{1}\,{\rm d}x_{2}\,{\rm d}t$
	$\displaystyle\leqslant$	$\displaystyle Cr^{\frac{3}{4}}+C\sup_{-1<t<0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{2}(B(1))}^{2}\cdot\left(\int_{-r^{2}}^{0}\int_{-\rho}^{\rho}\int_{B^{\prime}(r)}\|\nabla\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}.$

We can pick a sufficient small $0<R_{0}<1$ , such that for all $0<r<R_{0}$ ,

\displaystyle r^{-2}\left(\|\bm{u}\|_{L^{3}(Q(r))}^{3}+\|\bm{b}\|_{L^{3}(Q(r))}^{3}\right)<\varepsilon_{1}.

By Lemma A.9, we have that the solution is regular at $(0,0)$ , which leads to a contradiction.∎∎

Now, we are in position to prove Theorem 1.2 by standard rigid methods.

For $r_{k}=2^{-k},\ k\geqslant 1$ and $(x,t)\in Q(1)$ , we denote the following scaling

	$\displaystyle\bm{u}_{k}(x,t)=r_{k}\bm{u}\left(r_{k}x,r_{k}^{2}t\right),\ \bm{b}_{k}(x,t)=r_{k}\bm{b}\left(r_{k}x,r_{k}^{2}t\right)$
	$\displaystyle\pi_{k}(x,t)=r_{k}^{2}\pi\left(r_{k}x,r_{k}^{2}t\right).$

By (3.22), we obtain that for $k\geqslant 1$ ,

(4.8)

\displaystyle\sup_{t\in(-1,0)}\|(\bm{u}_{k},\bm{b}_{k})\|^{2}_{L^{2}(B(1))}+\|\nabla(\bm{u}_{k},\bm{b}_{k})\|^{2}_{L^{2}(Q(1))}\leqslant C.

Applying Lemma 3.1 and Lemma A.6, we obtain that for $0<r\leqslant 1$ ,

\displaystyle r^{-2}\|\pi\|_{L^{\frac{3}{2}}(Q(r))}^{\frac{3}{2}}\leqslant C,

which implies that

(4.9)

\displaystyle\|\pi_{k}\|_{L^{\frac{3}{2}}(Q(1))}\leqslant C.

Moreover, $\left(\bm{u}_{k},\bm{b}_{k},\pi_{k}\right)$ is a suitable weak solution to the MHD equations (1.1) in $Q(1)$ , which converges weakly (by taking subsequences if needed) to some $\left(\bm{v},\bm{B},\Pi\right)$ ,

(4.10)

\displaystyle\sup_{t\in(-1,0)}\|\left(\bm{v},\bm{B}\right)\|^{2}_{L^{2}(B(1))}+\|\nabla\left(\bm{v},\bm{B}\right)\|^{2}_{L^{2}(Q(1))}+\|\Pi\|_{L^{\frac{3}{2}}(Q(1))}\leqslant C.

By a similar argument as [17, Theorem 2.2], we can prove that $\left(\bm{v},\bm{B},\Pi\right)$ is a suitable weak solution to the MHD equations in $Q(1)$ and $\left(\bm{u}_{k},\bm{b}_{k}\right)\underset{k\rightarrow\infty}{\longrightarrow}\left(\bm{v},\bm{B}\right)$ strongly in $L^{3}(Q(1))$ . By (1.8)–(1.9), we have

(4.11)

\displaystyle v_{3}=B_{3}=0.

By Lemma 4.1, the solution $(\bm{v},\bm{B})$ is regular at $(0,0)$ . If we assume that the solution $(\bm{u},\bm{b})$ is singular at $(0,0)$ , $(\bm{u}_{k},\bm{b}_{k})$ is also singular at $(0,0)$ . It leads to a contradiction by Lemma A.8.

The proof of Theorem 1.2 is completed.∎

Appendix A

Lemma A.1.

[3, Lemma A.2] Let $0<r\leqslant R<+\infty$ and $h:B^{\prime}(2R)\times(-r,r)\rightarrow\mathbb{R}$ be harmonic. Then for all $0<\rho\leqslant\frac{r}{4}$ and $1\leqslant\ell\leqslant q<+\infty$ , we get

(A.1)

\displaystyle\|h\|_{L^{q}\left(B^{\prime}(R)\times(-\rho,\rho)\right)}^{q}\leqslant C\rho r^{2-\frac{3q}{\ell}}\|h\|_{L^{\ell}\left(B^{\prime}(2R)\times(-r,r)\right)}^{q},

where $C$ stands for a positive constant depending only on $q$ and $\ell$ .

We will present the estimates of $J$ , $K$ and $H$ , which is defined in (3.2), in the following several lemmas.

Lemma A.2.

Under the assumptions of Lemma 3.1, we have

(A.2)

\displaystyle J\leqslant C\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right)+C\mathcal{E}^{\frac{3}{2}}.

Proof.

Let $\pi_{0,j}=\mathscr{J}\left(\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\phi_{j}\right)$ .

	$\displaystyle J=$	$\displaystyle\sum_{k=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\sum_{k=4}^{n}\sum_{j=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\text{d}s$
		$\displaystyle+\sum_{k=0}^{3}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\text{d}s$
	$\displaystyle=$	$\displaystyle\sum_{k=4}^{n}\sum_{j=0}^{k-4}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\text{d}s$
		$\displaystyle+\sum_{k=4}^{n}\sum_{j=k-3}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\text{d}s$
		$\displaystyle+\sum_{k=0}^{3}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\text{d}s$
(A.3)		$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}$	$\displaystyle J_{1}+J_{2}+J_{3}.$

By (2.1), (3.14), Lemma 2.1 and Lemma A.1 with $\rho=r_{k},r=r_{j+2}$ , we have

	$\displaystyle J_{1}=$	$\displaystyle\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{-2}\int_{-r_{k}^{2}}^{0}\\|\pi_{0,j}\\|_{L^{\frac{q_{0}}{q_{0}-1}}(U_{k}(R))}\\|u_{3}\\|_{L^{q_{0}}(U_{k}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{-1-\frac{1}{q_{0}}}r_{j}^{-1-\frac{1}{2q_{0}}}\int_{-r_{k}^{2}}^{0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{4q_{0}}{2q_{0}-1}}(U_{j}(R))}^{2}\\|u_{3}\\|_{L^{q_{0}}(U_{j}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{1-\frac{2}{p_{0}^{*}}-\frac{5}{2q_{0}}}r_{j}^{-1-\frac{1}{2q_{0}}}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{8q_{0}}{3}}\left(-r_{j}^{2},0;L^{\frac{4q_{0}}{2q_{0}-1}}(U_{j}(R))\right)}^{2}$
		$\displaystyle\qquad\qquad\quad\times\\|u_{3}\\|_{L^{p_{0}^{*}}\left(-r_{j}^{2},0;L^{q_{0}}(U_{j}(R))\right)}$
(A.4)		$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right).$

By (2.1), (3.14) and Lemma 2.1, we have

	$\displaystyle J_{2}=$	$\displaystyle\sum_{k=4}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\mathscr{J}\left(\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\chi_{k-3}\right)\cdot u_{3}\cdot\partial_{3}\mathrm{\Phi}_{n}\phi_{k}\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=4}^{n}r_{k}^{-2}\int_{-r_{k}^{2}}^{0}\\|\mathscr{J}\left(\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\chi_{k-3}\right)\\|_{L^{\frac{q_{0}}{q_{0}-1}}(U_{k}(R))}\\|u_{3}\\|_{L^{q_{0}}(U_{k}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=0}^{n}r_{k}^{-2}\int_{-r_{k}^{2}}^{0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{\frac{2q_{0}}{q_{0}-1}}(U_{k}(R))}^{2}\\|u_{3}\\|_{L^{q_{0}}(U_{k}(R))}\,{\rm d}s$
(A.5)		$\displaystyle\lesssim$	$\displaystyle\sum_{i=0}^{n}\mathscr{B}_{i}\left(r_{i}^{-1}E_{i}(R)\right),$

which is analogous to (3.1). By (2.1) and (3.14), we have

(A.6)

\displaystyle J_{3}\lesssim\int_{-1}^{0}\|\pi_{0}\|_{L^{\frac{3}{2}}(U_{0}(R))}\|u_{3}\|_{L^{3}(U_{0}(R))}\,{\rm d}s\lesssim\int_{-1}^{0}\|\left(\bm{u},\bm{b}\right)\|_{L^{3}(U_{0}(R))}^{3}\,{\rm d}s\lesssim\mathcal{E}^{\frac{3}{2}}.

Summing up all the estimates of (A), (A), (A) and (A.6), we have (A.2).∎∎

Lemma A.3.

Under the assumptions of Lemma 3.1, we have

(A.7)

\displaystyle K\leqslant\frac{C\mathcal{E}^{\frac{1}{2}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right)+\frac{C}{R-\rho}\mathcal{E}^{\frac{3}{2}}.

Proof.

	$\displaystyle K=$	$\displaystyle\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\mathrm{\Phi}_{n}\eta\nabla\psi\,{\rm d}x\,{\rm d}s+\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\mathrm{\Phi}_{n}\nabla\eta\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s+\int_{-1}^{0}\int_{U_{0}(R)}\|\pi_{0}\|\|\bm{u}\|\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=4}^{n}\sum_{j=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\ \bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s$
		$\displaystyle+\sum_{k=0}^{3}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0}\bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s+\int_{-1}^{0}\int_{U_{0}(R)}\|\pi_{0}\|\|\bm{u}\|\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=4}^{n}\sum_{j=0}^{k-4}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\ \bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s$
		$\displaystyle+\sum_{k=4}^{n}\sum_{j=k-3}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\ \bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s+\frac{1}{R-\rho}\int_{-1}^{0}\int_{U_{0}(R)}\|\pi_{0}\|\|\bm{u}\|\,{\rm d}x\,{\rm d}s$
(A.8)		$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}$	$\displaystyle K_{1}+K_{2}+K_{3}.$

Analogously with (A), we have

	$\displaystyle K_{1}=$	$\displaystyle\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{0,j}\ \bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{R-\rho}\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{-1}\int_{-r_{k}^{2}}^{0}\\|\pi_{0,j}\\|_{L^{\frac{3}{2}}(U_{k}(R))}\\|\bm{u}\\|_{L^{3}(U_{k}(R))}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{R-\rho}\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{-\frac{1}{3}}r_{j}^{-\frac{2}{3}}\int_{-r_{k}^{2}}^{0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{3}(U_{j}(R))}^{3}\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{R-\rho}\sum_{j=0}^{n-4}\sum_{k=j+4}^{n}r_{k}^{\frac{1}{6}}r_{j}^{-\frac{2}{3}}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{4}\left(-r_{k}^{2},0;L^{3}(U_{j}(R))\right)}^{3}$
(A.9)		$\displaystyle\lesssim$	$\displaystyle\frac{\mathcal{E}^{\frac{1}{2}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right).$

Analogously with (A), we have

	$\displaystyle K_{2}=$	$\displaystyle\sum_{k=4}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\mathscr{J}\left(\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\chi_{k-3}\right)\ \bm{u}\cdot\mathrm{\Phi}_{n}\phi_{k}\nabla\psi\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{R-\rho}\sum_{k=0}^{n}r_{k}^{-1}\int_{-r_{k}^{2}}^{0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{3}(U_{k}(R))}^{3}\,{\rm d}s$
(A.10)		$\displaystyle\lesssim$	$\displaystyle\frac{\mathcal{E}^{\frac{1}{2}}}{R-\rho}\sum_{i=0}^{n}r_{i}^{\frac{1}{2}}\left(r_{i}^{-1}E_{i}(R)\right).$

Analogously with (A.6), we have

(A.11)

\displaystyle K_{3}\lesssim\frac{1}{R-\rho}\int_{-1}^{0}\|\left(\bm{u},\bm{b}\right)\|_{L^{3}(U_{0}(R))}^{3}\,{\rm d}s\lesssim\frac{1}{R-\rho}\mathcal{E}^{\frac{3}{2}}.

Summing up (A), (A), (A) and (A.11), we have (A.7).∎∎

Lemma A.4.

Under the assumptions of Lemma 3.1, we have

(A.12)

H\leqslant\frac{C}{(R-\rho)^{3}}\mathcal{E}^{\frac{3}{2}}.

Proof.

	$\displaystyle H=$	$\displaystyle-\sum_{k=0}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\nabla\pi_{h}\cdot\bm{u}\cdot\left(\mathrm{\Phi}_{n}\phi_{k}\psi\right)\,{\rm d}x\,{\rm d}s$
	$\displaystyle=$	$\displaystyle\sum_{k=0}^{3}\int_{-1}^{t}\int_{U_{0}(R)}\pi_{h}\cdot\bm{u}\cdot\nabla\left(\mathrm{\Phi}_{n}\phi_{k}\psi\right)\,{\rm d}x\,{\rm d}s-\sum_{k=4}^{n}\int_{-1}^{t}\int_{U_{0}(R)}\nabla\pi_{h}\cdot\bm{u}\cdot\left(\mathrm{\Phi}_{n}\phi_{k}\psi\right)\,{\rm d}x\,{\rm d}s$
(A.13)		$\displaystyle\buildrel\hbox{\footnotesize def}\over{=}$	$\displaystyle H_{1}+H_{2}.$

Applying (2.1), Hölder’s inequality and the fact that

(A.14)

\displaystyle\|\pi_{h}\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})}\leqslant

\displaystyle\|\pi\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})}+\|\pi_{0}\|_{L^{\frac{3}{2}}(\mathbb{R}^{3})}\lesssim\|\left(\bm{u},\bm{b}\right)\|_{L^{3}(\mathbb{R}^{3})}^{2},

we have

(A.15)

\displaystyle H_{1}\lesssim\frac{1}{R-\rho}\int_{-1}^{t}\int_{U_{0}(R)}|\pi_{h}||\bm{u}|\,{\rm d}x\,{\rm d}s\lesssim\frac{1}{R-\rho}\|\pi_{h}\|_{L^{\frac{3}{2}}\left(Q_{0}(1)\right)}\cdot\|\bm{u}\|_{L^{3}\left(Q_{0}(1)\right)}\lesssim\frac{1}{R-\rho}\mathcal{E}^{\frac{3}{2}}.

Moreover,

	$\displaystyle H_{2}\lesssim$	$\displaystyle\sum_{k=4}^{n}r_{k}^{-1}\int_{Q_{k}(\frac{R+\rho}{2})}\|\nabla\pi_{h}\|\cdot\|\bm{u}\|\,{\rm d}x\,{\rm d}s$
	$\displaystyle\lesssim$	$\displaystyle\sum_{k=4}^{n}r_{k}^{-1}\left\\|\nabla\pi_{h}\right\\|_{L^{\frac{3}{2}}\left(Q_{k}(\frac{R+\rho}{2})\right)}\\|\bm{u}\\|_{L^{3}\left(Q_{k}(R)\right)}$
(A.16)		$\displaystyle\lesssim$	$\displaystyle\sum_{k=4}^{n}r_{k}^{-\frac{1}{3}}\left\\|\nabla\pi_{h}\right\\|_{L^{\frac{3}{2}}\left(-r_{k}^{2},0;L^{\infty}\left(U_{k}(\frac{R+\rho}{2})\right)\right)}\\|\bm{u}\\|_{L^{3}\left(Q_{k}(R)\right)}.$

For any $x^{*}\in U_{k}(\frac{R+\rho}{2})$ , we have $B\left(x^{*},\frac{R-\rho}{4}\right)\subset U_{1}(R)$ due to $k\geqslant 4$ and $|R-\rho|\leqslant\frac{1}{2}.$ Since $\pi_{h}$ is harmonic in $U_{1}(R),$ using the mean value property, we have

\displaystyle\left|\nabla\pi_{h}\right|\left(x^{*}\right)\lesssim\frac{1}{|R-\rho|^{4}}\int_{B\left(x^{*},\frac{R-\rho}{4}\right)}|\pi_{h}|\,{\rm d}x\lesssim\frac{1}{(R-\rho)^{3}}\left\|\pi_{h}\right\|_{L^{\frac{3}{2}}\left(U_{1}(R)\right)}.

Hence,

	$\displaystyle H_{2}\lesssim$	$\displaystyle\frac{1}{(R-\rho)^{3}}\sum_{k=4}^{n}r_{k}^{-\frac{1}{3}}\left\\|\pi_{h}\right\\|_{L^{\frac{3}{2}}\left(-r_{k}^{2},0;L^{\frac{3}{2}}\left(U_{1}(R)\right)\right)}\\|\bm{u}\\|_{L^{3}\left(Q_{k}(R)\right)}$
	$\displaystyle\lesssim$	$\displaystyle\frac{1}{(R-\rho)^{3}}\sum_{k=4}^{n}r_{k}^{\frac{1}{6}}\ \\|\left(\bm{u},\bm{b}\right)\\|_{L^{4}\left(-r_{k}^{2},0;L^{3}\left(\mathbb{R}^{3}\right)\right)}^{3}$
(A.17)		$\displaystyle\lesssim$	$\displaystyle\frac{1}{(R-\rho)^{3}}\ \mathcal{E}^{\frac{3}{2}}.$

Summing up (A), (A.15) and (A) , we have (A.12).∎∎

Lemma A.5.

[24, Lemma A.3] For any

\displaystyle 1\leqslant p_{*}<p=\frac{2q}{q-3},\quad 3<q<+\infty,

we have

\displaystyle\sum_{k=0}^{+\infty}r_{k}^{1-\frac{2}{p_{*}}-\frac{3}{q}}\left(\int_{-r_{k}^{2}}^{0}\left\|f(\cdot,s)\right\|_{L^{q}(B(2))}^{p_{*}}~{}ds\right)^{\frac{1}{p_{*}}}\leqslant C\left\|f\right\|_{L^{p,1}\left(-1,0;L^{q}(B(2))\right)}.

For the sake of simplicity, we define

(A.18)		$\displaystyle D(\pi,z_{0},r)=r^{-2}\int_{Q(z_{0},r)}\left\|\pi\right\|^{\frac{3}{2}}\,{\rm d}x\,{\rm d}t,\ F(\bm{u},\bm{b},z_{0},r)=r^{-2}\int_{Q(z_{0},r)}\|\bm{u}\|^{3}+\|\bm{b}\|^{3}\,{\rm d}x\,{\rm d}t,$
(A.19)		$\displaystyle D(\pi,r)\buildrel\hbox{\footnotesize def}\over{=}D(\pi,(0,0),r),\ F(\bm{u},\bm{b},r)\buildrel\hbox{\footnotesize def}\over{=}C(\bm{u},\bm{b},(0,0),r).$

Lemma A.6.

Let $\pi\in L^{\frac{3}{2}}(Q(1))$ solve $\Delta\pi=\nabla\cdot\nabla\cdot\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)$ in the sense of distributions. If there exists a constant $K_{0}$ such that for all $0<r\leqslant R\leqslant 1$ ,

(A.20)

\displaystyle F(\bm{u},\bm{b},r)\leqslant K_{0},

then for some $\alpha>0$ and all $0<r\leqslant R$ ,

(A.21)

\displaystyle D(\pi,r)\leqslant C\left(\frac{r}{R}\right)^{\alpha}\cdot D(\pi,R)+CK_{0}.

Proof.

We claim that for $0<2r<\rho\leqslant R$ ,

(A.22)

\displaystyle D(\pi,r)\leq C_{2}\left(\frac{r}{\rho}\right)D(\pi,\rho)+C_{2}\left(\frac{\rho}{r}\right)^{2}F(\bm{u},\bm{b},\rho).

Actually, we write $\pi=\pi_{0}+\pi_{h}$ , where $\pi_{0}=\mathscr{J}\left(\left(\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\right)\cdot\chi_{B(\rho)}\right)$ and

(A.23)

\displaystyle D(\pi_{0},r)\lesssim r^{-2}\|\bm{u}\otimes\bm{u}-\bm{b}\otimes\bm{b}\|_{L^{\frac{3}{2}}(Q(\rho))}^{\frac{3}{2}}\lesssim\left(\frac{\rho}{r}\right)^{2}F(\bm{u},\bm{b},\rho).

$\pi_{h}$ is harmonic in $B(\rho)$ and the mean value property of $\pi_{h}$ implies that

(A.24)

\displaystyle D(\pi_{h},r)\lesssim r\int_{-r^{2}}^{0}\|\pi_{h}\|_{L^{\infty}(B(r))}^{\frac{3}{2}}\,{\rm d}t\lesssim\left(\frac{r}{\rho}\right)D(\pi_{h},\rho)\lesssim\left(\frac{r}{\rho}\right)\left(D(\pi,\rho)+D(\pi_{0},\rho)\right).

Summing up (A.23) and (A.24), we have (A.22).

Let $\theta\in(0,\frac{1}{2})$ . By (A.20) and (A.22), we have that for $0<r\leqslant R$ ,

(A.25)

\displaystyle D(\pi,\theta r)\leqslant C_{2}\cdot\theta\ D(\pi,r)+C_{2}K_{0}\ \theta^{-2}.

We choose $\theta$ such that $C_{2}\cdot\theta=\frac{1}{2}$ . For $\theta R<r\leq R$ , we have

(A.26)

\displaystyle D(\pi,r)\leqslant\theta^{-2}R^{-2}\int_{Q(R)}|\pi|^{\frac{3}{2}}\,{\rm d}x\,{\rm d}t.

This together with a standard iteration yields (A.21) with $\alpha=-\frac{\ln 2}{\ln\theta}>0$ . ∎∎

Lemma A.7.

[23, Theorem 1.1] There exists an absolute constant $\varepsilon_{0}>0$ with the following property. If $(\bm{u},\bm{b},\pi)$ is a suitable weak solution to the MHD equations in $Q(1)$ satisfying that for some $0<R\leqslant 1$ , $Q(z_{0},R)\subseteq Q(1)$ and

(A.27)

\displaystyle R^{-2}\int_{Q(z_{0},R)}\left(|\bm{u}|^{3}+|\bm{b}|^{3}+|\pi|^{\frac{3}{2}}\right)\,{\rm d}x\,{\rm d}t<\varepsilon_{0},

then the solution is regular at $z_{0}=(x_{0},t_{0})$ .

Lemma A.8 (stability of singularities).

Let $\left(\bm{u}_{k},\bm{b}_{k},\pi_{k}\right)$ be a sequence of suitable weak solutions to the MHD equations (1.1) in $Q(1)$ such that $(\bm{u}_{k},\bm{b}_{k})\rightarrow(\bm{v},\bm{B})$ in $L^{3}(Q(1))$ , $\pi_{k}\rightharpoonup\Pi$ in $L^{\frac{3}{2}}(Q(1))$ . Assume $(\bm{u}_{k},\bm{b}_{k})$ is singular at $z_{k}=(x_{k},t_{k})$ , $z_{k}\rightarrow(0,0)$ as $k\rightarrow\infty$ . Then $(\bm{v},\bm{B})$ is singular at $(0,0)$ .

Proof.

The proof is similar with [19, Lemma 2.1]. If $(\bm{v},\bm{B})$ is regular at $(0,0)$ , then there exists $\rho_{0}>0$ and for all $0<r<\rho_{0}$ ,

(A.28)

\displaystyle r^{-2}\int_{Q(r)}|\bm{v}|^{3}+|\bm{B}|^{3}\,{\rm d}x\,{\rm d}t\leqslant C\ r^{3}.

Since $(\bm{u}_{k},\bm{b}_{k})$ is singular at $z_{k}$ , by (A.18) and Lemma A.7, we have that for all $0<r<\frac{1}{2}$ ,

(A.29)

\displaystyle F(\bm{u}_{k},\bm{b}_{k},z_{k},r)+D(\pi_{k},z_{k},r)\geqslant\varepsilon_{0}.

For sufficient large $N=N(r)$ and all $k\geqslant N$ , we have $Q(z_{k},r)\in Q(2r)$ and

(A.30)

\displaystyle F(\bm{u}_{k},\bm{b}_{k},2r)+D(\pi_{k},2r)\geqslant\frac{\varepsilon_{0}}{4}.

Denote $\tilde{F}(r)=\underset{k\rightarrow\infty}{\limsup}\ F(\bm{u}_{k},\bm{b}_{k},r)$ and $\tilde{D}(r)=\underset{k\rightarrow\infty}{\limsup}\ D(\pi_{k},r)$ . For all $0<r<1$ , we have

(A.31)

\displaystyle\tilde{F}(r)+\tilde{D}(r)\geqslant\frac{\varepsilon_{0}}{4}.

By (A.28), we have that for all $0<r<\rho<\rho_{0}$ ,

(A.32)

\displaystyle\tilde{F}(r)=F(\bm{v},\bm{B},r)\leqslant Cr^{3}\leqslant C\rho^{3}.

Applying an analogous argument in Lemma A.6, we have that for all $r$ with $0<r<\rho$ ,

(A.33)

\displaystyle\tilde{D}(r)\leqslant C\left(\frac{r}{\rho}\right)^{\alpha}\cdot\tilde{D}(\rho)+C\rho^{3}.

Accordingly, we have for all $0<r<\rho<\rho_{0}$ ,

(A.34)

\displaystyle C_{3}\left(\frac{r}{\rho}\right)^{\alpha}\cdot\tilde{D}(\rho)+C_{3}\rho^{3}\geqslant\frac{\varepsilon_{0}}{4},

where the constant $C_{3}$ is independent of $r$ and $\rho$ . It leads to a contradiction if we let $r\rightarrow 0$ and then $\rho\rightarrow 0$ . The proof is completed. ∎∎

Lemma A.9.

[25, Theorem 1.1] There is an absolute number $\varepsilon_{1}>0$ with the following property. If $(\bm{u},\bm{b},\pi)$ is a suitable weak solution of (1.1) in $Q(1)$ satisfying that for some $0<R_{0}<1$ and all $0<r<R_{0}$ ,

(A.35)

\displaystyle r^{-2}\int_{Q(r)}|\bm{u}|^{3}\,{\rm d}x\,{\rm d}t<\varepsilon_{1},

then the solution $(\bm{u},\bm{b})$ is regular at $(0,0)$ .

Acknowledgments

Hui Chen was supported by Natural Science Foundation of Zhejiang Province(LQ19A010002). Chenyin Qian was supported by Natural Science Foundation of Zhejiang Province(LY20A010017). Ting Zhang was in part supported by National Natural Science Foundation of China (11771389, 11931010, 11621101).

References

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		$\displaystyle\frac{d}{dt}\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+4\int_{\mathbb{R}^{3}}\|\nabla\widetilde{\bm{Z}^{+}}\|^{2}\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\|^{2}\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}^{3}}\bm{Z}^{-}\cdot\nabla\alpha^{4}\cdot\beta^{4}\|\widetilde{\bm{Z}^{+}}\|^{4}-4\widetilde{\bm{Z}^{-}}\cdot\nabla\overline{\bm{Z}^{+}}\cdot\|\widetilde{\bm{Z}^{+}}\|^{2}\widetilde{\bm{Z}^{+}}\varphi^{4}\,{\rm d}x$
		$\displaystyle+4\int_{\mathbb{R}^{3}}\left(\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{-}}\cdot\nabla\widetilde{\bm{Z}^{+}}\cdot\beta-\bm{Z}^{+}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}\right)\cdot\|\widetilde{\bm{Z}^{+}}\|^{2}\widetilde{\bm{Z}^{+}}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\widetilde{\bm{Z}^{+}}\|^{4}\|\nabla\varphi^{2}\|^{2}\,{\rm d}x$
	$\displaystyle\lesssim$	$\displaystyle 1+\left(\\|\nabla\overline{\bm{Z}^{+}}\\|_{L^{2}(B^{\prime}(\rho))}+\\|\nabla\widetilde{\bm{Z}^{+}}\\|_{L^{2}(U(\rho))}\right)\cdot\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}$
		$\displaystyle\times\\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}+\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{3}$
	$\displaystyle\leqslant$	$\displaystyle C+C\left(1+\\|\nabla\bm{Z}^{+}\\|_{L^{2}(U(\rho))}^{2}\right)\cdot\left(\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}\right)$
(4.5)			$\displaystyle-\\|\nabla\left(\|\widetilde{\bm{Z}^{+}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}^{2}.$

		$\displaystyle\frac{d}{dt}\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}+4\int_{\mathbb{R}^{3}}\|\nabla\widetilde{\bm{Z}^{-}}\|^{2}\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\nabla\left(\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{2}\right)\|^{2}\,{\rm d}x$
	$\displaystyle=$	$\displaystyle\int_{\mathbb{R}^{3}}\bm{Z}^{+}\cdot\nabla\alpha^{4}\cdot\beta^{4}\|\widetilde{\bm{Z}^{-}}\|^{4}-4\widetilde{\bm{Z}^{+}}\cdot\nabla\overline{\bm{Z}^{-}}\cdot\|\widetilde{\bm{Z}^{-}}\|^{2}\widetilde{\bm{Z}^{-}}\varphi^{4}\,{\rm d}x$
		$\displaystyle+4\int_{\mathbb{R}^{3}}\left(\int_{-\rho}^{\rho}\widetilde{\bm{Z}^{+}}\cdot\nabla\widetilde{\bm{Z}^{-}}\cdot\beta-\bm{Z}^{-}\cdot\partial_{3}^{2}\beta\,{\rm d}x_{3}\right)\cdot\|\widetilde{\bm{Z}^{-}}\|^{2}\widetilde{\bm{Z}^{-}}\varphi^{4}\,{\rm d}x+2\int_{\mathbb{R}^{3}}\|\widetilde{\bm{Z}^{-}}\|^{4}\|\nabla\varphi^{2}\|^{2}\,{\rm d}x$
	$\displaystyle\leqslant$	$\displaystyle C+C\left(1+\\|\nabla\bm{Z}^{-}\\|_{L^{2}(U(\rho))}^{2}\right)\cdot\left(\\|\widetilde{\bm{Z}^{+}}\varphi\\|_{L^{4}(U(\rho))}^{4}+\\|\widetilde{\bm{Z}^{-}}\varphi\\|_{L^{4}(U(\rho))}^{4}\right)$
(4.6)			$\displaystyle-\\|\nabla\left(\|\widetilde{\bm{Z}^{-}}\|^{2}\varphi^{2}\right)\\|_{L^{2}(U(\rho))}^{2}.$

		$\displaystyle r^{-2}\left(\\|\bm{u}\\|_{L^{3}(Q(r))}^{3}+\\|\bm{b}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{-2}\left(\\|\bm{Z}^{+}\\|_{L^{3}(Q(r))}^{3}+\\|\bm{Z}^{-}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{-2}\left(\\|\widetilde{\bm{Z}^{+}}\\|_{L^{3}(Q(r))}^{3}+\\|\widetilde{\bm{Z}^{-}}\\|_{L^{3}(Q(r))}^{3}\right)+Cr^{-2}\left(\\|\overline{\bm{Z}^{+}}\\|_{L^{3}(Q(r))}^{3}+\\|\overline{\bm{Z}^{-}}\\|_{L^{3}(Q(r))}^{3}\right)$
	$\displaystyle\leqslant$	$\displaystyle Cr^{\frac{3}{4}}+Cr^{-1}\int_{-r^{2}}^{0}\int_{B^{\prime}(r)}\left(\|\overline{\bm{Z}^{+}}\|^{3}+\|\overline{\bm{Z}^{-}}\|^{3}\right)\,{\rm d}x_{1}\,{\rm d}x_{2}\,{\rm d}t$
	$\displaystyle\leqslant$	$\displaystyle Cr^{\frac{3}{4}}+C\sup_{-1<t<0}\\|\left(\bm{u},\bm{b}\right)\\|_{L^{2}(B(1))}^{2}\cdot\left(\int_{-r^{2}}^{0}\int_{-\rho}^{\rho}\int_{B^{\prime}(r)}\|\nabla\left(\bm{u},\bm{b}\right)\|^{2}\,{\rm d}x\,{\rm d}t\right)^{\frac{1}{2}}.$