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

Abstract

In this paper, we consider the 3D MHD equations in which the viscosity coefficient $\mu$ and the resistivity coefficient $\nu$ are equal. 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

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

$\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1$ and $3<q_{0},q_{1}<+\infty$, imply that the suitable weak solution is regular at (0, 0).

Appendix

Lemma A.1

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

$$\begin{aligned} \Vert h\Vert _{L^{q}\left( B^{\prime }(R) \times (-\rho , \rho )\right) }^{q} \leqslant C \rho r^{2- \frac{3q}{\ell }}\Vert h\Vert _{L^{\ell }\left( B^{\prime }(2 R) \times (-r, r)\right) }^{q}, \end{aligned}$$

(A.1)

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 (3.16), in the following several lemmas.

Lemma A.2

Under the assumptions of Lemma 3.1, we have

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

(A.2)

Proof

Let $\pi _{0,j}={\mathscr {J}}\left( \left( \varvec{u}\otimes \varvec{u}-\varvec{b}\otimes \varvec{b}\right) \cdot \phi _{j}\right)$.

$$\begin{aligned} J=&\sum _{k=0}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ =&\sum _{k=4}^{n}\sum _{j=0}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\text {d}s \nonumber \\&+\sum _{k=0}^{3}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\text {d}s \nonumber \\ =&\sum _{k=4}^{n}\sum _{j=0}^{k-4}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\text {d}s \nonumber \\&+\sum _{k=4}^{n}\sum _{j=k-3}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\text {d}s \nonumber \\&+\sum _{k=0}^{3}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\text {d}s \nonumber \\ \buildrel \hbox { def}\over =&J_{1}+J_{2}+J_{3}. \end{aligned}$$

(A.3)

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

$$\begin{aligned} J_{1} =&\sum _{j=0}^{n-4}\sum _{k=j+4}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j}\cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\sum _{j=0}^{n-4}\sum _{k=j+4}^{n} r_{k}^{-2}\int _{-r_{k}^2}^{0}\Vert \pi _{0,j}\Vert _{L^{\frac{q_{0}}{q_{0}-1}}(U_{k}(R))}\Vert u_{3}\Vert _{L^{q_{0}}(U_{k}(R))} \,\mathrm{d}s\nonumber \\ \lesssim&\sum _{j=0}^{n-4}\sum _{k=j+4}^{n} r_{k}^{-1-\frac{1}{q_{0}}}r_{j}^{-1-\frac{1}{2q_{0}}}\int _{-r_{k}^2}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{\frac{4q_{0}}{2q_{0}-1}}(U_{j}(R))}^{2}\Vert u_{3}\Vert _{L^{q_{0}}(U_{j}(R))} \,\mathrm{d}s\nonumber \\ \lesssim&\sum _{j=0}^{n-4}\sum _{k=j+4}^{n} r_{k}^{1-\frac{2}{p_{0}^*}-\frac{5}{2q_{0}}}r_{j}^{-1-\frac{1}{2q_{0}}}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{\frac{8q_{0}}{3}}\left( -r_{j}^2,0;L^{\frac{4q_{0}}{2q_{0}-1}}(U_{j}(R))\right) }^2\nonumber \\&\times \Vert u_{3}\Vert _{L^{p_{0}^*}\left( -r_{j}^2,0;L^{q_{0}}(U_{j}(R))\right) }\nonumber \\ \lesssim&\sum _{i=0}^{n} \mathscr {B}_{i} \left( r_{i}^{-1} E_{i}(R)\right) . \end{aligned}$$

(A.4)

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

$$\begin{aligned} J_{2}=&\sum _{k=4}^{n}\int _{-1}^{t} \int _{U_{0}(R)} {\mathscr {J}}\left( \left( \varvec{u}\otimes \varvec{u}-\varvec{b}\otimes \varvec{b}\right) \cdot \chi _{k-3}\right) \cdot u_{3} \cdot \partial _{3}{\Phi }_{n} \phi _{k} \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\sum _{k=4}^{n}r_{k}^{-2}\int _{-r_{k}^2}^{0}\Vert {\mathscr {J}}\left( \left( \varvec{u}\otimes \varvec{u}-\varvec{b}\otimes \varvec{b}\right) \cdot \chi _{k-3}\right) \Vert _{L^{\frac{q_{0}}{q_{0}-1}}(U_{k}(R))}\Vert u_{3}\Vert _{L^{q_{0}}(U_{k}(R))} \,\mathrm{d}s\nonumber \\ \lesssim&\sum _{k=0}^{n} r_{k}^{-2}\int _{-r_{k}^2}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{\frac{2q_{0}}{q_{0}-1}}(U_{k}(R))}^{2} \Vert u_{3}\Vert _{L^{q_{0}}(U_{k}(R))} \,\mathrm{d}s\nonumber \\ \lesssim&\sum _{i=0}^{n} \mathscr {B}_{i} \left( r_{i}^{-1} E_{i}(R)\right) , \end{aligned}$$

(A.5)

which is analogous to (3.10). By (2.1) and (3.14), we get

$$\begin{aligned} J_{3}\lesssim \int _{-1}^{0}\Vert \pi _{0}\Vert _{L^{\frac{3}{2}}(U_{0}(R))} \Vert u_{3}\Vert _{L^{3}(U_{0}(R))} \,\mathrm{d}s\lesssim \int _{-1}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{3}(U_{0}(R))}^3 \,\mathrm{d}s\lesssim \mathcal {E}^{\frac{3}{2}}. \end{aligned}$$

(A.6)

Summing up all the estimates of (A.3), (A.4), (A.5) and (A.6), we obtain (A.2).$\square$

Lemma A.3

Under the assumptions of Lemma 3.1, we have

$$\begin{aligned} K\leqslant \frac{C\mathcal {E}^{\frac{1}{2}}}{R-\rho } \sum _{i=0}^{n} r_{i}^{\frac{1}{2}}\left( r_{i}^{-1} E_{i}(R)\right) +\frac{C}{R-\rho }\mathcal {E}^{\frac{3}{2}}. \end{aligned}$$

(A.7)

Proof

$$\begin{aligned} K=&\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0} \varvec{u} \cdot {\Phi }_{n} \eta \nabla \psi \,\mathrm{d}x\,\mathrm{d}s+\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0} \varvec{u} \cdot {\Phi }_{n} \nabla \eta \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\sum _{k=0}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0} \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s+\int _{-1}^{0}\int _{U_{0}(R)}|\pi _{0}| |\varvec{u}|\,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\sum _{k=4}^{n}\sum _{j=0}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j} \ \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\&+\sum _{k=0}^{3}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0} \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s+\int _{-1}^{0}\int _{U_{0}(R)}|\pi _{0}||\varvec{u}|\,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\sum _{k=4}^{n}\sum _{j=0}^{k-4}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j} \ \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\&+\sum _{k=4}^{n}\sum _{j=k-3}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j} \ \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s+\frac{1}{R-\rho }\int _{-1}^{0}\int _{U_{0}(R)}|\pi _{0}||\varvec{u}|\,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \buildrel \hbox { def}\over =&K_{1}+K_{2}+K_{3}. \end{aligned}$$

(A.8)

Analogously with (A.4), we obtain

$$\begin{aligned} K_{1}=&\sum _{j=0}^{n-4}\sum _{k=j+4}^{n}\int _{-1}^{t} \int _{U_{0}(R)} \pi _{0,j} \ \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\frac{1}{R-\rho }\sum _{j=0}^{n-4}\sum _{k=j+4}^{n}r_{k}^{-1}\int _{-r_{k}^2}^{0}\Vert \pi _{0,j}\Vert _{L^{\frac{3}{2}}(U_{k}(R))}\Vert \varvec{u}\Vert _{L^{3}(U_{k}(R))}\,\mathrm{d}s\nonumber \\ \lesssim&\frac{1}{R-\rho }\sum _{j=0}^{n-4}\sum _{k=j+4}^{n}r_{k}^{-\frac{1}{3}}r_{j}^{-\frac{2}{3}}\int _{-r_{k}^2}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{3}(U_{j}(R))}^{3}\,\mathrm{d}s\nonumber \\ \lesssim&\frac{1}{R-\rho }\sum _{j=0}^{n-4}\sum _{k=j+4}^{n}r_{k}^{\frac{1}{6}}r_{j}^{-\frac{2}{3}}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{4}\left( -r_{k}^2,0;L^{3}(U_{j}(R))\right) }^{3}\nonumber \\ \lesssim&\frac{\mathcal {E}^{\frac{1}{2}}}{R-\rho } \sum _{i=0}^{n} r_{i}^{\frac{1}{2}}\left( r_{i}^{-1} E_{i}(R)\right) . \end{aligned}$$

(A.9)

Analogously with (A.5), we get

$$\begin{aligned} K_{2}=&\sum _{k=4}^{n}\int _{-1}^{t} \int _{U_{0}(R)} {\mathscr {J}}\left( \left( \varvec{u}\otimes \varvec{u}-\varvec{b}\otimes \varvec{b}\right) \cdot \chi _{k-3}\right) \ \varvec{u} \cdot {\Phi }_{n} \phi _{k} \nabla \psi \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \lesssim&\frac{1}{R-\rho }\sum _{k=0}^{n}r_{k}^{-1}\int _{-r_{k}^2}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{3}(U_{k}(R))}^3\,\mathrm{d}s\nonumber \\ \lesssim&\frac{\mathcal {E}^{\frac{1}{2}}}{R-\rho } \sum _{i=0}^{n} r_{i}^{\frac{1}{2}}\left( r_{i}^{-1} E_{i}(R)\right) . \end{aligned}$$

(A.10)

Analogously with (A.6), we have

$$\begin{aligned} K_{3}\lesssim \frac{1}{R-\rho }\int _{-1}^{0}\Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{3}(U_{0}(R))}^3 \,\mathrm{d}s\lesssim \frac{1}{R-\rho }\mathcal {E}^{\frac{3}{2}}. \end{aligned}$$

(A.11)

Summing up (A.8), (A.9), (A.10) and (A.11), we obtain (A.7).$\square$

Lemma A.4

Under the assumptions of Lemma 3.1, we have

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

(A.12)

Proof

$$\begin{aligned} H=&-\sum _{k=0}^{n}\int _{-1}^{t}\int _{U_{0}(R)} \nabla \pi _{h}\cdot \varvec{u}\cdot \left( {\Phi }_{n} \phi _{k} \psi \right) \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ =&\sum _{k=0}^{3}\int _{-1}^{t}\int _{U_{0}(R)} \pi _{h}\cdot \varvec{u}\cdot \nabla \left( {\Phi }_{n} \phi _{k} \psi \right) \,\mathrm{d}x\,\mathrm{d}s-\sum _{k=4}^{n}\int _{-1}^{t}\int _{U_{0}(R)} \nabla \pi _{h}\cdot \varvec{u}\cdot \left( {\Phi }_{n} \phi _{k} \psi \right) \,\mathrm{d}x\,\mathrm{d}s\nonumber \\ \buildrel \hbox { def}\over =&H_{1}+H_{2}. \end{aligned}$$

(A.13)

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

$$\begin{aligned} \Vert \pi _{h}\Vert _{L^{\frac{3}{2}}({\mathbb {R}}^3)}\leqslant&\Vert \pi \Vert _{L^{\frac{3}{2}}({\mathbb {R}}^3)}+\Vert \pi _{0}\Vert _{L^{\frac{3}{2}}({\mathbb {R}}^3)}\lesssim \Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{3}({\mathbb {R}}^3)}^2, \end{aligned}$$

(A.14)

we obtain

$$\begin{aligned} H_{1}\lesssim \frac{1}{R-\rho }\int _{-1}^{t}\int _{U_{0}(R)}|\pi _{h}||\varvec{u}|\,\mathrm{d}x\,\mathrm{d}s\lesssim \frac{1}{R-\rho }\Vert \pi _{h}\Vert _{L^{\frac{3}{2}}\left( Q_{0}(1)\right) }\Vert \varvec{u}\Vert _{L^{3}\left( Q_{0}(1)\right) } \lesssim \frac{1}{R-\rho } \mathcal {E}^{\frac{3}{2}}. \end{aligned}$$

(A.15)

Moreover,

$$\begin{aligned} H_{2}\lesssim & {} \sum _{k=4}^{n} r_{k}^{-1} \int _{Q_{k}(\frac{R+\rho }{2}) }|\nabla \pi _{h}| \cdot |\varvec{u}|\,\mathrm{d}x\,\mathrm{d}s\nonumber \\\lesssim & {} \sum _{k=4}^{n} r_{k}^{-1}\left\| \nabla \pi _{h}\right\| _{L^{\frac{3}{2}}\left( Q_{k}(\frac{R+\rho }{2}) \right) }\Vert \varvec{u}\Vert _{L^{3}\left( Q_{k}(R) \right) }\nonumber \\\lesssim & {} \sum _{k=4}^{n} r_{k}^{-\frac{1}{3}}\left\| \nabla \pi _{h}\right\| _{L^{\frac{3}{2}}\left( -r_{k}^{2}, 0 ; L^{\infty }\left( U_{k}(\frac{R+\rho }{2}) \right) \right) }\Vert \varvec{u}\Vert _{L^{3}\left( Q_{k}(R) \right) }. \end{aligned}$$

(A.16)

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

$$\begin{aligned} \left| \nabla \pi _{h}\right| \left( x^{*}\right) \lesssim \frac{1}{|R-\rho |^{4}} \int _{B\left( x^*, \frac{R-\rho }{4}\right) }|\pi _{h}|\,\mathrm{d}x\lesssim \frac{1}{(R-\rho )^{3}}\left\| \pi _{h}\right\| _{L^ \frac{3}{2}\left( U_{1}(R) \right) }. \end{aligned}$$

Hence,

$$\begin{aligned} H_{2}\lesssim&\frac{1}{(R-\rho )^{3}} \sum _{k=4}^{n} r_{k}^{-\frac{1}{3}}\left\| \pi _{h}\right\| _{L^{\frac{3}{2}}\left( -r_{k}^{2}, 0 ; L^{\frac{3}{2}}\left( U_{1}(R)\right) \right) }\Vert \varvec{u}\Vert _{L^{3}\left( Q_{k}(R)\right) } \nonumber \\ \lesssim&\frac{1}{(R-\rho )^{3}} \sum _{k=4}^{n} r_{k}^{\frac{1}{6}}\ \Vert \left( \varvec{u},\varvec{b}\right) \Vert _{L^{4}\left( -r_{k}^{2}, 0 ; L^{3}\left( {\mathbb {R}}^3\right) \right) }^{3}\nonumber \\ \lesssim&\frac{1}{(R-\rho )^{3}}\ \mathcal {E}^{\frac{3}{2}}. \end{aligned}$$

(A.17)

Summing up (A.13), (A.15) and (A.17), we get (A.12).$\square$

Lemma A.5

([24], Lemma A.3) For any

$$\begin{aligned} 1 \leqslant p_*<p=\frac{2q}{q-3}, \quad 3<q<+\infty , \end{aligned}$$

we have

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

For the sake of simplicity, we define

$$\begin{aligned}&D(\pi ,z_{0},r)=r^{-2}\int _{Q(z_{0},r)}\left| \pi \right| ^{\frac{3}{2}}\,\mathrm{d}x\,\mathrm{d}t,\ F(\varvec{u},\varvec{b},z_{0},r)=r^{-2}\int _{Q(z_{0},r)}|\varvec{u}|^3+|\varvec{b}|^3\,\mathrm{d}x\,\mathrm{d}t, \end{aligned}$$

(A.18)

$$\begin{aligned}&D(\pi ,r)\buildrel \hbox { def}\over =D(\pi ,(0,0),r), \ F(\varvec{u},\varvec{b},r)\buildrel \hbox { def}\over =C(\varvec{u},\varvec{b},(0,0),r). \end{aligned}$$

(A.19)

Lemma A.6

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

$$\begin{aligned} F(\varvec{u},\varvec{b},r)\leqslant K_{0}, \end{aligned}$$

(A.20)

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

$$\begin{aligned} D(\pi ,r) \leqslant C \left( \frac{r}{R}\right) ^{\alpha }\cdot D(\pi ,R)+C K_{0}. \end{aligned}$$

(A.21)

Proof

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

$$\begin{aligned} D(\pi ,r)\le C_{2}\left( \frac{r}{\rho }\right) D(\pi ,\rho )+C_{2}\left( \frac{\rho }{r}\right) ^2 F(\varvec{u},\varvec{b},\rho ). \end{aligned}$$

(A.22)

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

$$\begin{aligned} D(\pi _{0},r)\lesssim r^{-2}\Vert \varvec{u}\otimes \varvec{u}-\varvec{b}\otimes \varvec{b}\Vert _{L^{\frac{3}{2}}(Q(\rho ))}^\frac{3}{2}\lesssim \left( \frac{\rho }{r}\right) ^2 F(\varvec{u},\varvec{b},\rho ). \end{aligned}$$

(A.23)

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

$$\begin{aligned} D(\pi _{h},r)\lesssim r \int _{-r^2}^{0}\Vert \pi _{h}\Vert _{L^{\infty }(B(r))}^{\frac{3}{2}}\,\mathrm{d}t\lesssim \left( \frac{r}{\rho }\right) D(\pi _{h},\rho )\lesssim \left( \frac{r}{\rho }\right) \left( D(\pi ,\rho )+D(\pi _{0},\rho )\right) . \end{aligned}$$

(A.24)

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 obtain that for $0<r\leqslant R$,

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

(A.25)

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

$$\begin{aligned} D(\pi ,r)\leqslant \theta ^{-2} R^{-2}\int _{Q(R)}|\pi |^{\frac{3}{2}}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$

(A.26)

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

Lemma A.7

([23], Theorem 1.1) There exists an absolute constant $\varepsilon _{0}>0$ with the following property. If $(\varvec{u},\varvec{b},\pi )$ 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

$$\begin{aligned} R^{-2} \int _{Q(z_{0},R)}\left( |\varvec{u}|^{3}+|\varvec{b}|^{3}+|\pi |^{\frac{3}{2}}\right) \,\mathrm{d}x\,\mathrm{d}t<\varepsilon _{0}, \end{aligned}$$

(A.27)

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

Lemma A.8

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

Proof

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

$$\begin{aligned} r^{-2}\int _{Q(r)}|\varvec{v}|^3+|\varvec{B}|^3\,\mathrm{d}x\,\mathrm{d}t\leqslant C \ r^3. \end{aligned}$$

(A.28)

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

$$\begin{aligned} F(\varvec{u}_{k},\varvec{b}_{k},z_{k},r)+D(\pi _{k},z_{k},r)\geqslant \varepsilon _{0}. \end{aligned}$$

(A.29)

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

$$\begin{aligned} F(\varvec{u}_{k},\varvec{b}_{k},2r)+D(\pi _{k},2r)\geqslant \frac{\varepsilon _{0}}{4}. \end{aligned}$$

(A.30)

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

$$\begin{aligned} \tilde{F}(r)+\tilde{D}(r)\geqslant \frac{\varepsilon _{0}}{4}. \end{aligned}$$

(A.31)

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

$$\begin{aligned} \tilde{F}(r)=F(\varvec{v},\varvec{B},r)\leqslant Cr^3\leqslant C \rho ^3. \end{aligned}$$

(A.32)

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

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

(A.33)

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

$$\begin{aligned} C_{3} \left( \frac{r}{\rho }\right) ^{\alpha }\cdot \tilde{D}(\rho )+C_{3} \rho ^3 \geqslant \frac{\varepsilon _{0}}{4}, \end{aligned}$$

(A.34)

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. $\square$

Lemma A.9

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

$$\begin{aligned} r^{-2} \int _{Q(r)}|\varvec{u}|^{3}\,\mathrm{d}x\,\mathrm{d}t< \varepsilon _{1}, \end{aligned}$$

(A.35)

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