Regularity of Weak Solutions to the 3D Magneto-Micropolar Equations in Besov Spaces

Abstract

This paper deals with the regularity of weak solutions to the 3D magneto-micropolar fluid equations in Besov spaces. It is shown that for $0\le\alpha\le1$ if $u\in L^{\frac{2}{1+\alpha}}(0,T; \dot{B}_{\infty,\infty}^{\alpha})$, then the weak solution $(u,\omega ,b)$ is regular on $(0,T]$.

Appendix

Theorems 1.4 and 1.6 were proved in [44] and [37], we give their proof here for completeness.

Proof of Theorem 1.4

We derive the following estimate by differentiating (3.1) and multiplying $\partial_{x_{i}} u$, $\partial_{x_{i}} \omega$, $\partial_{x_{i}} b$, respectively,

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\partial_{x_{i}} u\|_{2}^{2}+ \|\partial_{x_{i}} \omega\|_{2}^{2}+\|\partial_{x_{i}} b\|_{2}^{2}\bigr)+ \sum_{j=1}^{3}(\mu+\chi)\bigl\| \partial_{x_{i}x_{j}}^{2} u\bigr\| _{2}^{2}+\sum_{j=1}^{3}\gamma\bigl\| \partial_{x_{i}x_{j}}^{2} \omega \bigr\| _{2}^{2} \\ &\qquad{}+\sum_{j=1}^{3}\nu\bigl\| \partial_{x_{i}x_{j}}^{2} b\bigr\| _{2}^{2} +\sum_{j=1}^{3}\kappa\|\operatorname{div}\partial_{x_{i}}\omega\|_{2}^{2}+\sum_{j=1}^{3}2\chi\|\partial_{x_{i}} \omega\|_{2}^{2} \\ &\quad\le\bigl|(\partial_{x_{i}} u\cdot\nabla u,\partial_{x_{i}}u)\bigr|+ \bigl|(\partial_{x_{i}} b\cdot\nabla b,\partial_{x_{i}} u)\bigr|+ \bigl|(\partial_{x_{i}} u\cdot\nabla b,\partial_{x_{i}} b)\bigr| \\ &\qquad{}+\bigl|(\partial_{x_{i}} b\cdot\nabla u,\partial_{x_{i}} b)\bigr|+ \bigl|(\partial_{x_{i}} u\cdot\nabla \omega,\partial_{x_{i}} \omega)\bigr|+ 2\chi\bigl|\bigl(\nabla\times\omega,\partial_{x_{i}x_{j}}^{2} u\bigr)\bigr| \\ &\quad\triangleq I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}. \end{aligned}$$

(A.1)

We use the Littlewood-Paley decomposition to decompose $u$

$$\begin{aligned} u=\sum_{j\in\mathbb{Z}}\Delta_{j}u= \sum_{j< -N}\Delta_{j}u+\sum_{|j|\le N}\Delta_{j}u+ \sum_{j>N}\Delta_{j}u, \end{aligned}$$

(A.2)

where $N$is a positive integer that will be determined later. Inserting (A.2) into $I _{1}$ one has

$$\begin{aligned} I_{1}&= \sum_{j< -N}\int_{\mathbb{R}^{3}}\partial_{x_{i}} u\cdot\nabla u\Delta_{j} \partial_{x_{i}}u\mathrm{d}x +\sum_{|j|\le N}\int_{\mathbb{R}^{3}}\partial_{x_{i}} u\cdot\nabla u\Delta_{j}\partial_{x_{i}}u\mathrm{d}x \\ &\qquad{} + \sum_{j>N}\int_{\mathbb{R}^{3}}\partial_{x_{i}} u\cdot\nabla u\Delta_{j}\partial_{x_{i}}u\mathrm{d}x \\ &\quad=I_{11}+I_{12}+I_{13}. \end{aligned}$$

(A.3)

By virtue of the Hölder inequality and Bernstein inequality we obtain that

$$\begin{gathered} |I_{11}|\le\|\nabla u\|_{2}^{2}\sum_{j< -N}\|\Delta_{j}\partial_{x_{i}}u\|_{\infty}\le C2^{-\frac{3}{2}N}\|\nabla u\|_{2}^{3}, \end{gathered}$$

(A.4)

$$\begin{gathered} |I_{12}|\le\|\nabla u\|_{2}^{2}\sum_{|j|\le N}\|\Delta_{j}\nabla u\|_{\infty}\le 3N\|\nabla u\|_{2}^{2}\|\nabla u\|_{\dot{B}_{\infty,\infty}^{0}}\le 3N\|\nabla u\|_{2}^{2}\|\nabla u\|_{\dot{B}_{\infty,\infty}^{1}}. \end{gathered}$$

(A.5)

For $I_{13}$, by the Gagliardo-Nirenberg inequality, one also has

$$\begin{aligned} |I_{13}|\le\sum_{j>n}2^{j}\|\Delta_{j}\nabla u\|_{2}2^{-\frac{j}{2}}\|\nabla u\|_{3}^{2}\le 2^{-\frac{N}{2}}\|\nabla u\|_{2}\bigl\| D^{2} u\bigr\| _{2}^{2}. \end{aligned}$$

(A.6)

Summing up estimates (A.4)–(A.6), it follows

$$\begin{aligned} |I_{1}|\le C2^{-\frac{3}{2}N}\|\nabla u\|_{2}^{3}+3N\|\nabla u\|_{2}^{2}\|\nabla u\|_{\dot{B}_{\infty,\infty}^{0}}+ 2^{-\frac{N}{2}}\|\nabla u\|_{2}\bigl\| D^{2} u\bigr\| _{2}^{2}. \end{aligned}$$

(A.7)

Similar to the estimate of $I _{1}$, the term $I _{2},\ldots ,I _{5}$ can be bounded as

$$\begin{gathered} \begin{aligned}[b] |I _{2}|,\ldots,|I _{4}|&\le C2^{-\frac{3}{2}N}\bigl(\|\nabla b\|_{3}^{2}+\|\nabla u\|_{3}^{2}\bigr)+ CN\|\nabla b\|_{2}^{2}\| u\|_{\dot{B}_{\infty,\infty}^{1}}\\ &\quad{}+C2^{-\frac{N}{2}}\|\nabla b\|_{2} \bigl(\bigl\| D^{2} u\bigr\| _{2}^{2}+\bigl\| D^{2}b\bigr\| _{2}^{2}\bigr), \end{aligned} \end{gathered}$$

(A.8)

$$\begin{gathered} \begin{aligned}[b] |I_{5}|&\le C2^{-\frac{3}{2}N}\bigl(\|\nabla \omega\|_{3}^{2}+\|\nabla u\|_{3}^{2}\bigr)+ CN\|\nabla \omega\|_{2}^{2}\|u\|_{\dot{B}_{\infty,\infty}^{1}}\\ &\quad{}+ C2^{-\frac{N}{2}}\|\nabla \omega\|_{2} \bigl(\bigl\| D^{2} u\bigr\| _{2}^{2}+\bigl\| D^{2}\omega\bigr\| _{2}^{2}\bigr). \end{aligned} \end{gathered}$$

(A.9)

Substituting (A.7)–(A.9) and (3.9) into (A.1), and summing up $i$ from 1 to 3, it follows that

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\nabla u\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+\|\nabla b\|_{2}^{2} \bigr)+ \biggl(\mu+\frac{\chi}{2}\biggr)\bigl\| D^{2} u\bigr\| _{2}^{2} +\gamma\bigl\| D^{2}\omega\|_{2}^{2} +\nu\bigl\| D^{2} b\bigr\| _{2}^{2} \\ &\qquad{}+\kappa\|\operatorname{div}\nabla\omega\|_{2}^{2} \\ &\quad\le C2^{-\frac{3N}{2}}\bigl(\|\nabla u\|_{2}^{3}+\|\nabla b\|_{2}^{3}+ \|\nabla \omega\|_{2}^{3}\bigr)+CN \bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}\bigr)\|u\|_{\dot{B}_{\infty,\infty}^{1}} \\ &\qquad{} + C2^{-\frac{N}{2}}\bigl(\|\nabla u\|_{2}+\|\nabla b\|_{2}+ \|\nabla \omega\|_{2}\bigr) \bigl(\bigl\| D^{2} u\bigr\| _{2}^{2}+\bigl\| D^{2}b\bigr\| _{2}^{2}+\bigl\| D^{2}\omega\bigr\| _{2}^{2}\bigr). \end{aligned}$$

(A.10)

We choose

$$\begin{aligned} N\ge\biggl[\frac{2}{\log2}\log\frac{C(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+e)}{\min(\mu+\frac{\chi}{2},\gamma,\nu)}\biggr]+2. \end{aligned}$$

Thus (A.10) can be reduced to

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\nabla u\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+\|\nabla b\|_{2}^{2} \bigr)+ (\mu+\chi)\bigl\| D^{2} u\bigr\| _{2}^{2} +\gamma\bigl\| D^{2}\omega\|_{2}^{2} +\nu\bigl\| D^{2} b\bigr\| _{2}^{2} \\ &\qquad{}+\kappa\|\operatorname{div}\nabla\omega\|_{2}^{2} \\ &\quad\le C +C \bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}\bigr)\|u\|_{\dot{B}_{\infty,\infty}^{1}} \bigl(\log\bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+e\bigr)\bigr). \end{aligned}$$

Applying the Gronwall inequality gives that for $t \in [0,T)$

$$\begin{aligned} &\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+\|\nabla \omega\|_{2}^{2} \\ &\quad\le \bigl(\|\nabla u_{0}(x)\|_{2}^{2}+\|\nabla b_{0}(x)\|_{2}^{2}+ \|\nabla \omega_{0}(x)\|_{2}^{2}+CT\bigr) \\ &\qquad{}\times\exp\biggl(\int_{0}^{t}C\bigl(\| u\|_{\dot{B}_{\infty,\infty}^{1}}\bigr) \log\bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+e\bigr)\mathrm{d}\tau\biggr). \end{aligned}$$

(A.11)

Taking logarithmic on both sides of (A.11) and applying the Gronwall inequality to it, we obtain that

$$\begin{aligned} &\log\bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+e\bigr) \\ &\quad\le \log\bigl(\|\nabla u_{0}(x)\|_{2}^{2}+\|\nabla b_{0}(x)\|_{2}^{2}+ \|\nabla \omega_{0}(x)\|_{2}^{2}+CT\bigr)\exp \biggl(\int_{0}^{t}\| u\|_{\dot{B}_{\infty,\infty}^{1}}\mathrm{d}\tau\biggr), \end{aligned}$$

(A.12)

for any $t \in [0,T)$. Therefore, by the standard arguments of continuation of local solutions, we complete the proof of Theorem 1.4. □

Proof of Theorem 1.6

In this section we prove Theorem 1.6 by energy methods. If (1.6) holds, one can deduce that for any small constant $> 0$, there exists $T _{0} < T$ such that

$$\begin{aligned} \int_{T_{0}}^{T}\| u\|_{\dot{B}_{\infty,\infty}^{0}}\mathrm{d}\tau\le\epsilon. \end{aligned}$$

(A.13)

We adopt the idea that was first used in [23] to prove Theorem 1.6. We first derive the $L ^{2}$ and $H ^{1}$ estimates by $\int_{T_{0}}^{t}\|u\|_{\mathit{BMO}}^{2}\mathrm{d}\tau$. And then by the logarithmic Sobolev inequality (2.10) we improve it by (A.13).

Step I. $L ^{2}$ and $H ^{1}$-estimates.

First, taking the $L ^{2}$ inner product of (1.1) with $u$, $b$ and $\omega$, respectively. It follows that

$$\begin{aligned} &\bigl\| (u,\omega,b)\bigr\| _{2}^{2}+ 2\mu\int_{\varepsilon}^{t}\|\nabla u\|_{2}^{2}\mathrm{d}s+ 2\gamma\int_{\varepsilon}^{t}\|\nabla \omega\|_{2}^{2}\mathrm{d}s+ 2\nu\int_{\varepsilon}^{t}\|\nabla b\|_{2}^{2}\mathrm{d}s \\ &\quad{}+ 2\kappa\int_{\varepsilon}^{t}\|\operatorname{div}\omega\|_{2}^{2}\mathrm{d}s+ 2\chi\int_{\varepsilon}^{t}\|\omega\|_{2}^{2}\mathrm{d}s\le \bigl\| \bigl(u(T_{0}),\omega(T_{0}),b(T_{0})\bigr)\bigr\| _{2}^{2} \end{aligned}$$

(A.14)

for $0 \leq T _{0}\leq t \leq T$.

Second, similar to the proof of Theorem 1.1, we start this step from the term in (3.1). This time we estimate $I _{1},\ldots ,I _{6}$ in another way. By Lemma 2.3 and Young inequality, we derive the following estimates

$$\begin{aligned} I_{1}&=\biggl|\int_{\mathbb{R}^{3}}u_{k}\partial_{i}(\partial_{k}u_{j}\partial_{i}u_{j})\mathrm{d}x\biggr| \\ &\le C\|u\|_{\mathit{BMO}}\|\nabla u\|_{2}\|\Delta u\|_{2} \\ &\le\frac{\mu}{2}\|\Delta u\|_{2}^{2}+C\|u\|_{\mathit{BMO}}^{2}\|\nabla u\|_{2}^{2}. \end{aligned}$$

(A.15)

Arguing similarly as the above discussion, one has

$$\begin{gathered} \begin{aligned}[b] I_{2},I_{3},I_{4}&=\biggl|\int_{\mathbb{R}^{3}}\partial_{i}(\partial_{i}b_{k}\partial_{k}b_{j})u_{j}\mathrm{d}x\biggr|\\ &\le \|u\|_{\mathit{BMO}}\|\nabla b\|_{2}\|\Delta b\|_{2}\\ &\le\frac{\nu}{6}\|\Delta b\|_{2}^{2}+C\|u\|_{\mathit{BMO}}^{2}\|\nabla b\|_{2}^{2}, \end{aligned} \end{gathered}$$

(A.16)

$$\begin{gathered} I_{5}\le\frac{\gamma}{2}\|\Delta \omega\|_{2}^{2}+C\|u\|_{\mathit{BMO}}^{2}\|\nabla \omega\|_{2}^{2}. \end{gathered}$$

(A.17)

Substituting (A.15)–(A.17) and (3.9) into (3.1), one has

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\nabla u\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+\|\nabla b\|_{2}^{2} \bigr)+ \biggl(\frac{\mu+\chi}{2}\biggr)\|\Delta u\|_{2}^{2} +\frac{\gamma}{2}\|\Delta\omega\|_{2}^{2} +\frac{\nu}{2}\|\Delta b\|_{2}^{2} \\ &\qquad{}+\kappa\|\operatorname{div}\nabla\omega\|_{2}^{2} \\ &\quad\le C\bigl(\|\nabla u\|_{2}^{2}+\|\nabla \omega\|_{2}^{2}+ \|\nabla b\|_{2}^{2}\bigr) \|u\|_{\mathit{BMO}}^{2}. \end{aligned}$$

(A.18)

The Gronwall inequality implies that

$$\begin{aligned} &\|\nabla u\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ (\mu+\chi)\int_{T_{0}}^{t}\|\Delta u\|_{2}^{2}\mathrm{d}\tau+ \gamma\int_{T_{0}}^{t}\|\Delta \omega\|_{2}^{2}\mathrm{d}\tau \\ &\qquad{}+ \nu\int_{T_{0}}^{t}\|\Delta b\|_{2}^{2}\mathrm{d}\tau+ 2\kappa\int_{T_{0}}^{t}\|\operatorname{div}\nabla\omega\|_{2}^{2}\mathrm{d}\tau \\ &\quad\le \bigl(\bigl\| \nabla u(T_{0})\bigr\| _{2}^{2}+ \bigl\| \nabla \omega(T_{0})\bigr\| _{2}^{2}+\bigl\| \nabla b(T_{0})\bigr\| _{2}^{2}\bigr)\exp \biggl(C\int_{T_{0}}^{t}\| u\|_{\mathit{BMO}}^{2}\mathrm{d}\tau\biggr). \end{aligned}$$

(A.19)

For any $t \in [T _{0} ,T]$, we denote

$$\begin{aligned} G(t)=\sup_{T_{0}\le \tau\le t}\bigl(\bigl\| u(\tau,\cdot)\bigr\| _{H^{2}}^{2}+ \bigl\| \omega(\tau,\cdot)\bigr\| _{H^{2}}^{2}+ \bigl\| b(\tau,\cdot)\bigr\| _{H^{2}}^{2}\bigr). \end{aligned}$$

(A.20)

Applying the logarithmic Sobolev inequality (2.10) to (A.19), we have

$$\begin{aligned} &\|\nabla u\|_{2}^{2}+ \|\nabla \omega\|_{2}^{2}+\|\nabla b\|_{2}^{2}+ (\mu+\chi)\int_{T_{0}}^{t}\|\Delta u\|_{2}^{2}\mathrm{d}\tau+ \gamma\int_{T_{0}}^{t}\|\Delta \omega\|_{2}^{2}\mathrm{d}\tau \\ &\qquad{}+ \nu\int_{T_{0}}^{t}\|\Delta b\|_{2}^{2}\mathrm{d}\tau+ 2\kappa\int_{T_{0}}^{t}\|\operatorname{div}\nabla\omega\|_{2}^{2}\mathrm{d}\tau \\ &\quad\le C(T_{0}) \exp \biggl(C\int_{T_{0}}^{t}\| u\|_{\dot{B}_{\infty,\infty}^{0}}^{2}\log^{++} \bigl(\|u\|_{H^{2}}^{2}+\|\omega\|_{H^{2}}^{2}+\|b\|_{H^{2}}^{2}\bigr)\mathrm{d}\tau\biggr) \\ &\quad\le C(T_{0})G(t)^{C_{0}\epsilon}. \end{aligned}$$

(A.21)

Step II. $H ^{2}$-estimates.

Applying operator $\Delta$ in (1.1) and taking the $L ^{2}$ inner product of the resulting equations with $\Delta u$, $\Delta \omega$ and $\Delta b$, respectively, we obtain

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\Delta u\|_{2}^{2}+ \|\Delta\omega\|_{2}^{2}+\|\Delta b\|_{2}^{2} \bigr)+ (\mu+\chi)\bigl\| \nabla^{3} u\bigr\| _{2}^{2}+\gamma\bigl\| \nabla^{3} \omega \bigr\| _{2}^{2} +\nu\bigl\| \nabla^{3} b\bigr\| _{2}^{2} \\ &\qquad{}+\kappa\|\operatorname{div}\Delta\omega\|_{2}^{2}+2\chi\|\Delta \omega\|_{2}^{2} \\ &\quad\le\bigl|(\Delta u\cdot\nabla u,\Delta u)\bigr| +2\bigl|(\partial_{i}u\cdot\nabla\partial_{i}u,\Delta u)\bigr| +\bigl|(\Delta b\cdot\nabla b,\Delta u)\bigr| \\ &\qquad{} +2\bigl|(\partial_{i}b\cdot\nabla\partial_{i}b,\Delta u)\bigr| +\bigl|(\Delta u\cdot\nabla b,\Delta b)\bigr| +2\bigl|(\partial_{i}u\cdot\nabla\partial_{i}b,\Delta b)\bigr| \\ &\qquad{} +\bigl|(\Delta b\cdot\nabla u,\Delta b)\bigr| +2\bigl|(\partial_{i}b\cdot\nabla\partial_{i}u,\Delta b)\bigr| + \bigl|(\Delta u\cdot\nabla \omega,\Delta \omega)\bigr| \\ &\qquad{} +2\bigl|(\partial_{i}u\cdot\nabla\partial_{i}\omega,\Delta \omega)\bigr| + 2\chi\bigl|(\nabla\times\Delta u,\Delta\omega)\bigr| \\ &\quad\triangleq \sum_{j=1}^{11}J_{j}. \end{aligned}$$

(A.22)

By Hölder, Gagliardo-Nirenberg and Young inequalities, we have

$$\begin{gathered} \begin{aligned}[b] J_{1},J_{2}&\le\|\nabla u\|_{2}\|\Delta u\|_{1}^{2} \\ &\le C\|\nabla u\|_{2}\|\Delta u\|_{2}^{\frac{1}{4}}\bigl\| \nabla^{3} u\bigr\| _{2}^{\frac{\gamma}{4}} \\ &\le\frac{\mu}{16} \bigl\| \nabla^{3} u\bigr\| _{2}^{2}+C\|\nabla u\|_{2}^{8}\|\nabla u\|_{2}^{2}, \end{aligned} \end{gathered}$$

(A.23)

$$\begin{gathered} \begin{aligned}[b] J_{3},J_{4},J_{5},J_{6},J_{7},J_{8}&\le \|\nabla b\|_{2}\bigl(\|\Delta u\|_{4}^{2}+\|\Delta b\|_{4}^{2}\bigr) \\ &\le \frac{\mu}{16}\bigl\| \nabla^{3} u\bigr\| _{2}^{2}+\frac{\nu}{12}\bigl\| \nabla^{3} b\bigr\| _{2}^{2} +C\|\nabla b\|_{2}^{8} \bigl(\|\nabla u\|_{2}^{2}+\|\nabla b\|_{2}^{2}\bigr). \end{aligned} \end{gathered}$$

(A.24)

Arguing similarly as the above, we obtain

$$\begin{aligned} J_{9},J_{10}\le \frac{\mu}{16}\bigl\| \nabla^{3} u\bigr\| _{2}^{2}+\frac{\gamma}{4}\bigl\| \nabla^{3} \omega\bigr\| _{2}^{2} +C\|\nabla \omega\|_{2}^{8} \bigl(\|\nabla u\|_{2}^{2}+\|\nabla \omega\|_{2}^{2}\bigr). \end{aligned}$$

(A.25)

We also have

$$\begin{aligned} J_{11}\le2\chi\|\Delta\omega\|_{2}^{2}+\frac{\chi}{2}\bigl\| \nabla^{3} u\bigr\| _{2}^{2}. \end{aligned}$$

(A.26)

Collecting (A.23)–(A.26) into (A.22), we arrive at

$$\begin{aligned} &\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \bigl(\|\Delta u\|_{2}^{2}+ \|\Delta \omega\|_{2}^{2}+\|\Delta b\|_{2}^{2} \bigr)+ \biggl(\frac{\mu+\chi}{2}\biggr) \bigl\| \nabla^{3} u\bigr\| _{2}^{2} +\frac{\gamma}{2}\bigl\| \nabla^{3} \omega\bigr\| _{2}^{2} \\ &\qquad{} +\frac{\nu}{2}\bigl\| \nabla^{3} b\bigr\| _{2}^{2} +\kappa\|\operatorname{div}\Delta\omega\|_{2}^{2} \\ &\quad\le C\bigl(\|\nabla u\|_{2}^{8}+\|\nabla \omega\|_{2}^{8}+ \|\nabla b\|_{2}^{8}\bigr) \bigl(\|\nabla u\|_{2}^{2}+\|\nabla \omega\|_{2}^{2}+ \|\nabla b\|_{2}^{2}\bigr) \\ &\quad\le C(T_{0})G(t)^{5C_{0}\epsilon}. \end{aligned}$$

(A.27)

Integrating the above estimate and combining (A.14), it can be derived that

$$\begin{aligned} G(t)\le \bigl\| \nabla^{2} u(T_{0})\bigr\| _{2}^{2}+ \bigl\| \nabla^{2} \omega(T_{0})\bigr\| _{2}^{2}+\bigl\| \nabla^{2} b(T_{0})\bigr\| _{2}^{2} + C\int_{T_{0}}^{t}G(\tau)^{5C_{0}\epsilon}\mathrm{d}\tau. \end{aligned}$$

(A.28)

Choosing small enough such that $5C_{0}\epsilon= 1$, it can be derived that

$$\begin{aligned} G(t) \leq C(T_{0}) < \infty . \end{aligned}$$

(A.29)

By the standard arguments of weak solution’s regularity, we complete the proof of Theorem 1.6. □