Appendix A
In this section, as mentioned, we give a proof of Lemma 4.1.
Proof of Lemma 4.1
We suppose that u is regular. We then compute certain a priori estimates. First of all, we note that by the \(L^2\)-energy estimate with (1.3),
$$\begin{aligned} \frac{d}{\text {d}t}\Vert u\Vert ^2_{L^2({\mathbb {R}}^3)}+m_0\Vert Du\Vert ^2_{L^2({\mathbb {R}}^3)}\le C\Vert u\Vert ^2_{L^2({\mathbb {R}}^3)}+C\Vert F\Vert ^2_{L^2({\mathbb {R}}^3)}. \end{aligned}$$
(A.71)
\(\bullet \) (\(\Vert \nabla u \Vert _{L^2}\)-estimate) Taking derivative \(\partial _{x_i}\) to (4.56) and multiplying \(\partial _{x_i}u\),
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_i}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_i} (G\left[ |Du|^2 \right] Du): \partial _{x_i} Du\,\text {d}x \\{} & {} \quad =- \mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_i} \big ((U\cdot \nabla ) u \big )\cdot \partial _{x_i} u\,\text {d}x-\mathop {\int }\limits _{{\mathbb {R}}^3} F \cdot \partial _{x_i}\partial _{x_i} u\,\text {d}x. \end{aligned}$$
Noting that
$$\begin{aligned} \begin{aligned} \partial _{x_i}(G\left[ |Du|^2 \right] Du): \partial _{x_i} Du&=\big [ \partial _{x_i}G\left[ |Du|^2 \right] Du + G\left[ |Du|^2 \right] \partial _{x_i} Du \big ]: \partial _{x_i} Du\\&= 2 G^{'}[|Du|^2](D u: \partial _{x_i}Du)( Du: \partial _{x_i} Du) + G\left[ |Du|^2 \right] |\partial _{x_i} Du|^2\\&= 2 G^{'}[|Du|^2]| Du: \partial _{x_i} Du|^2 + G\left[ |Du|^2 \right] |\partial _{x_i} Du|^2, \end{aligned} \end{aligned}$$
we have
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_i}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}G\left[ |Du|^2 \right] |\partial _{x_i} Du|^2\,\text {d}x + \mathop {\int }\limits _{{\mathbb {R}}^3}2G^{'}[|Du|^2]| Du: \partial _{x_i} Du|^2\,\text {d}x \nonumber \\{} & {} \quad = -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_i} \big ((U\cdot \nabla ) u \big )\cdot \partial _{x_i} u\,\text {d}x-\mathop {\int }\limits _{{\mathbb {R}}^3} F \cdot \partial _{x_i}\partial _{x_i} u\,\text {d}x. \end{aligned}$$
(A.72)
Taking \(A=Du\) and \(B=\partial _{x_i} D u\) in the last inequality, we use the following inequality:
$$\begin{aligned} G[|A|^2]|B|^2 + 2 G^{\prime }[|A|^2](A:B)^2 \ge m_0|B|^2. \end{aligned}$$
(A.73)
Indeed, if \(G^{\prime }[|A|^2] \ge 0\), then
$$\begin{aligned} G[|A|^2]|B|^2 + 2 G^{\prime }[|A|^2](A:B)^2 \ge G[|A|^2]|B|^2 \ge m_0|B|^2. \end{aligned}$$
In case that \(G^{\prime }[|A|^2] <0\), we note that
$$\begin{aligned} G[|A|^2]|B|^2 + 2 G^{\prime }[|A|^2](A:B)^2 \ge \big (G[|A|^2] + 2 G^{\prime }[|A|^2]|A|^2 \big )|B|^2\ge m_0 |B|^2. \end{aligned}$$
Applying inequality (A.73) to (A.72), we obtain
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_i}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}m_0 |\partial _{x_i} Du|^2\,\text {d}x \\&\quad \le -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_i} \big ((U\cdot \nabla ) u \big )\cdot \partial _{x_i} u\,\text {d}x-\mathop {\int }\limits _{{\mathbb {R}}^3} F \cdot \partial _{x_i}\partial _{x_i} u\,\text {d}x. \end{aligned} \end{aligned}$$
(A.74)
We will treat the term in right-hand side caused by convection together later.
\(\bullet \) (\(\Vert \nabla ^2 u \Vert _{L^2}\)-estimate) Taking the derivative \(\partial _{x_{j}}\partial _{x_{i}}\) on (4.56) and multiplying it by \(\partial _{x_{j}}\partial _{x_{i}}u \),
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_{j}}\partial _{x_{i}}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}\Big [G\left[ |Du|^2 \right] D u\Big ]: \partial _{x_{j}}\partial _{x_{i}}D u\,\text {d}x \nonumber \\{} & {} \quad = -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}\big ((U\cdot \nabla u) \big )\cdot \partial _{x_{j}}\partial _{x_{i}} u\,\text {d}x-\mathop {\int }\limits _{{\mathbb {R}}^3} \partial _{x_i}\partial _{x_i}F: \partial _{x_i}\partial _{x_i} u\,\text {d}x. \end{aligned}$$
(A.75)
We observe that
$$\begin{aligned} \begin{aligned}&\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}\Big [G\left[ |Du|^2 \right] D u\Big ]:\partial _{x_{j}}\partial _{x_{i}}D u\,\text {d}x\\&\quad =\mathop {\int }\limits _{{\mathbb {R}}^3}G\left[ |Du|^2 \right] |\partial _{x_{j}}\partial _{x_{i}}D u|^2\, +\sum _{\sigma }\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{\sigma (i)}}G\left[ |Du|^2 \right] (\partial _{x_{\sigma (j)}}D u: \partial _{x_{j}}\partial _{x_{i}}D u)\,\text {d}x\\&\qquad +\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}G\left[ |Du|^2 \right] (D u: \partial _{x_{j}}\partial _{x_{i}}D u)\,\text {d}x=:I_{21}+I_{22}+I_{23}, \end{aligned} \end{aligned}$$
(A.76)
where \(\sigma :\{ i,j\}\rightarrow \{i,j \}\) is a permutation of \(\{i,j \}\). We separately estimate terms \(I_{22}\) and \(I_{23}\) in (A.76). Using Hölder, Young’s and Gagliardo-Nirenberg inequalities, we have for \(I_{22}\)
$$\begin{aligned} |I_{22}|= & {} \left| \mathop {\int }\limits _{{\mathbb {R}}^3} 2G^{'}[|Du|^2](Du: \partial _{x_{\sigma (i)}}D u)(\partial _{x_{\sigma (j)}}D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,\text {d}x \right| \\\le & {} C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert \nabla Du\Vert ^2_{L^4}\Vert \nabla ^2 Du\Vert _{L^2} \\\le & {} C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert Du\Vert _{L^\infty }\Vert \nabla ^2 Du\Vert ^2_{L^2}, \end{aligned}$$
where we used condition (1.3).
For \(I_{23}\), using Lemma 2.3, we compute
$$\begin{aligned} \begin{aligned} I_{23}&= \mathop {\int }\limits _{{\mathbb {R}}^3}2\big ( G^{'}[|Du|^2](Du: \partial _{x_{j}}\partial _{x_{i}}Du) + E_2 \big ) (D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,\text {d}x\\&=\mathop {\int }\limits _{{\mathbb {R}}^3}E_2(D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,\text {d}x + 2\mathop {\int }\limits _{{\mathbb {R}}^3}G^{'}[|Du|^2] |Du: \partial _{x_{j}}\partial _{x_{i}}Du|^2\,\text {d}x\\&:=I_{231}+I_{232}. \end{aligned} \end{aligned}$$
The term \(I_{231}\) is estimated as
$$\begin{aligned} \begin{aligned} \left| I_{231} \right|&\le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty } \Vert Du\Vert _{L^\infty }\Vert \nabla Du\Vert ^2_{L^4}\Vert \nabla ^2 Du\Vert _{L^2}\\&\le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty } \Vert Du\Vert _{L^\infty }^2\Vert \nabla ^2 Du\Vert _{L^2}^2, \end{aligned} \end{aligned}$$
(A.77)
where we used the first inequality of (2.14). We combine estimates (A.75)–(A.77) to get
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_{j}}\partial _{x_{i}}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}G\left[ |Du|^2 \right] |\partial _{x_{j}}\partial _{x_{i}}D u|^2 + \mathop {\int }\limits _{{\mathbb {R}}^3}2G^{'}[|Du|^2]|Du: \partial _{x_{j}}\partial _{x_{i}}Du|^2 \\{} & {} \quad \le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }(\Vert Du\Vert _{L^\infty } + \Vert Du\Vert _{L^\infty }^2)\Vert \nabla ^2 Du\Vert ^2_{L^2} -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}\big ((u\cdot \nabla u) \big )\cdot \partial _{x_{j}}\partial _{x_{i}} u. \end{aligned}$$
Similarly as in (A.74), we have
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial _{x_{j}}\partial _{x_{i}}u||^2_{L^2({\mathbb {R}}^3)}+\mathop {\int }\limits _{{\mathbb {R}}^3}m_0|\partial _{x_{j}}\partial _{x_{i}}D u|^2 \nonumber \\{} & {} \quad \le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }(\Vert Du\Vert _{L^\infty } + \Vert Du\Vert _{L^\infty }^2)\Vert \nabla ^2 Du\Vert ^2_{L^2} \nonumber \\{} & {} \qquad -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{j}}\partial _{x_{i}}\big ((U\cdot \nabla u) \big )\cdot \partial _{x_{j}}\partial _{x_{i}} u-\mathop {\int }\limits _{{\mathbb {R}}^3} \partial _{x_i}F \cdot \partial _{x_j}\partial _{x_j}\partial _{x_i} u\,\text {d}x. \end{aligned}$$
(A.78)
\(\bullet \) (\(\Vert \nabla ^3 u \Vert _{L^2}\)-estimate) For convenience, we denote \(\partial ^3:=\partial _{x_{k}}\partial _{x_{j}}\partial _{x_{i}}\). Similarly as before, taking the derivative \(\partial ^3\) on (4.56) and multiplying it by \(\partial ^3 u\),
$$\begin{aligned}{} & {} \frac{1}{2}\frac{d}{\text {d}t}||\partial ^3u||^2_{L^2({\mathbb {R}}^3)} +\mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3\Big [G\left[ |Du|^2 \right] D u\Big ]: \partial ^3D u\,\text {d}x \nonumber \\{} & {} \quad -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3\big ((U\cdot \nabla u) \big )\cdot \partial ^3 u\,\text {d}x- \mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3F:\partial ^3 u\,\text {d}x. \end{aligned}$$
(A.79)
Direct computations show that
$$\begin{aligned}{} & {} \mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3\Big [G\left[ |Du|^2 \right] D u\Big ]: \partial ^3D u\,\text {d}x\\{} & {} \quad =\mathop {\int }\limits _{{\mathbb {R}}^3}G\left[ |Du|^2 \right] |\partial ^3D u|^2\,\text {d}x \\{} & {} \qquad +\sum _{\sigma _3}\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{\sigma _3(i)}}G\left[ |Du|^2 \right] (\partial _{x_{\sigma _3(k)}}\partial _{x_{\sigma _3(j)}}D u:\partial ^3D u)\,\text {d}x \\{} & {} \qquad +\sum _{\sigma _3}\mathop {\int }\limits _{{\mathbb {R}}^3}\partial _{x_{\sigma _3(j)}}\partial _{x_{\sigma _3(i)}}G\left[ |Du|^2 \right] (\partial _{x_{\sigma _3(k)}}D u: \partial ^3Du)\,\text {d}x \\{} & {} \qquad +\mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3G\left[ |Du|^2 \right] (D u:\partial ^3D u)\,\text {d}x=I_{31}+I_{32}+I_{33}+I_{34}, \end{aligned}$$
where \(\sigma _3=\pi _3\circ {\tilde{\sigma }}_3\) such that \({\tilde{\sigma }}_3:\{i,j,k\}\rightarrow \{i,j,k\}\) is a permutation of \(\{i,j,k \}\) and \(\pi _3\) is a mapping from \(\{i,j,k\}\) to \(\{1,2,3\}\).
We separately estimate terms \(I_{32}\), \(I_{33}\) and \(I_{34}\). We note first that
$$\begin{aligned} \begin{aligned} |I_{32}|&\le \mathop {\int }\limits _{{\mathbb {R}}^3}|2(G^{'}[|Du|^2]| |Du||\partial _{x_{\sigma _3(i)}}Du||\partial _{x_{\sigma _3(k)}}\partial _{x_{\sigma _3(j)}} D u| |\partial ^3 D u|\,\text {d}x\\&\le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert \nabla Du\Vert _{L^6}\Vert \nabla ^2 Du\Vert _{L^3}\Vert \nabla ^3 Du\Vert _{L^2}\\&\le C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert \nabla ^2 Du\Vert _{L^2} \Vert Du\Vert _{L^\infty }^{\frac{1}{3}} \Vert \nabla ^3 Du\Vert ^{\frac{2}{3}}_{L^2}\Vert \nabla ^3Du\Vert _{L^2}\\&\le C\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty }\Vert Du\Vert _{L^\infty }^2 \Vert \nabla ^2 Du\Vert ^{6}_{L^2}+\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2}. \end{aligned} \end{aligned}$$
(A.80)
For \(I_{33}\), we have
$$\begin{aligned} |I_{33}|= & {} \left| \mathop {\int }\limits _{{\mathbb {R}}^3}( 2G^{'}[|Du|^2]Du:\partial _{x_{\sigma _3(j)}} \partial _{x_{\sigma _3(i)}} D u + E_2 )(\partial _{x_{\sigma _3(k)}}D u: \partial ^3Du)\,\text {d}x \right| \nonumber \\\le & {} \mathop {\int }\limits _{{\mathbb {R}}^3}2(G^{'}[|Du|^2]|Du||\nabla ^2 D u| + G\left[ |Du|^2 \right] |\nabla D u|^2 )|\nabla D u| |\nabla ^3 Du|\,\text {d}x \nonumber \\\le & {} C \Vert G^{'}[|Du|^2]Du\Vert _{L^\infty } \Vert \nabla ^2 Du\Vert _{L^3}\Vert \nabla Du\Vert _{L^6}\Vert \nabla ^3 Du\Vert _{L^2} +\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert \nabla Du\Vert ^3_{L^6}\Vert \nabla ^3 Du\Vert _{L^2} \nonumber \\\le & {} C\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty }\Vert Du \Vert _{L^\infty }^2\Vert \nabla ^2 Du\Vert ^{6}_{L^2} +C\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }\Vert \nabla ^2Du\Vert ^6_{L^2}+2\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2} \nonumber \\\le & {} C(\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty }\Vert Du \Vert _{L^\infty }^2+\Vert G(|Du|)\Vert ^2_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}+2\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2}, \end{aligned}$$
(A.81)
where we use same argument as (A.80) in the fourth inequality. Finally, for \(I_{34}\), using Lemma 2.3, we note that
$$\begin{aligned} \begin{aligned} I_{34}=&\mathop {\int }\limits _{{\mathbb {R}}^3}\big ( 2G^{'}[|Du|^2]Du:\partial ^3 D u) + E_3\big )(D u:\partial ^3D u)\,\text {d}x\\ =&2\mathop {\int }\limits _{{\mathbb {R}}^3} G^{'}[|Du|^2]|Du:\partial ^3 D u|^2\,\text {d}x + \mathop {\int }\limits _{{\mathbb {R}}^3} E_3(D u:\partial ^3D u)\,\text {d}x. \end{aligned} \end{aligned}$$
(A.82)
The second term in (A.82) is estimated as follows:
$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^3}E_3(D u:\partial ^3D u)\,\text {d}x\le & {} \mathop {\int }\limits _{{\mathbb {R}}^3}|E_3||D u||\nabla ^3 D u|\,\text {d}x \nonumber \\\le & {} C\mathop {\int }\limits _{{\mathbb {R}}^3} G\left[ |Du|^2 \right] \big (|\nabla Du|^3 +|\nabla ^2 Du||\nabla Du| \big )|D u||\nabla ^3 D u|\,\text {d}x \nonumber \\\le & {} C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }\Vert Du\Vert _{L^\infty } \big ( \Vert \nabla Du\Vert _{L^6}^3 + \Vert \nabla Du\Vert _{L^6}\Vert \nabla ^2 Du\Vert _{L^3} \big )\Vert \nabla ^3 D u\Vert _{L^2} \nonumber \\\le & {} C\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }\Vert Du\Vert ^2_{L^\infty }\Vert \nabla ^2 Du\Vert ^6_{L^2}\nonumber \\{} & {} +C\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty }\Vert Du\Vert ^4_{L^\infty }\Vert \nabla ^2 Du\Vert ^{6}_{L^2}+2\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2} \nonumber \\\le & {} C(\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }\Vert Du\Vert ^2_{L^\infty }\nonumber \\{} & {} +\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty }\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}+2\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2}, \end{aligned}$$
(A.83)
where we use same argument as (A.81) in the third inequality. Adding up estimates (A.79)–(A.83), we obtain
$$\begin{aligned}{} & {} \frac{d}{\text {d}t}||\partial ^3u||^2_{L^2({\mathbb {R}}^3)} +\mathop {\int }\limits _{{\mathbb {R}}^3}G\left[ |Du|^2 \right] |\partial ^3D u|^2\,\text {d}x +\mathop {\int }\limits _{{\mathbb {R}}^3}2G^{'}[|Du|^2]|Du: \partial ^3Du|^2\,\text {d}x \\{} & {} \quad \le C(\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }+ \Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty } )(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}+5\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2} \\{} & {} \qquad -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3\big ((U\cdot \nabla u) \big )\cdot \partial ^3u\,\text {d}x- \mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^2F:\partial ^4 u\,\text {d}x. \end{aligned}$$
Hence, we have
$$\begin{aligned}{} & {} \frac{d}{\text {d}t}||\partial ^3u||^2_{L^2({\mathbb {R}}^3)} +\mathop {\int }\limits _{{\mathbb {R}}^3}m_0|\partial ^3D u|^2\,\text {d}x \nonumber \\{} & {} \quad \le C(\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }+ \Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty } )(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}+5\epsilon \Vert \nabla ^3 Du\Vert ^{2}_{L^2} \nonumber \\{} & {} \qquad -\mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^3\big ((U\cdot \nabla u) \big )\cdot \partial ^3u\,\text {d}x- \mathop {\int }\limits _{{\mathbb {R}}^3}\partial ^2F:\partial ^4 u\,\text {d}x. \end{aligned}$$
(A.84)
Next, we estimate the terms caused by convection terms in (A.74), (A.78) and (A.84).
$$\begin{aligned} \begin{aligned} \sum _{1\le |\alpha |\le 3}\mathop {\int }\limits _{{\mathbb {R}}^3} \partial ^{\alpha }[(U\cdot \nabla )u]\cdot \partial ^{\alpha }u\,\text {d}x&=\sum _{1\le |\alpha |\le 3}\mathop {\int }\limits _{{\mathbb {R}}^3} [\partial ^{\alpha }((U\cdot \nabla )u)-U\cdot \nabla \partial ^{\alpha }u]\partial ^{\alpha }u\,\text {d}x\\&\le \sum _{1\le |\alpha |\le 3}\Vert \partial ^{\alpha }((U\cdot \nabla )u)-U\cdot \nabla \partial ^{\alpha }u\Vert _{L^2}\Vert \partial ^{\alpha } u\Vert _{L^{2}}\\&\le \sum _{1\le |\alpha |\le 3} \Vert \nabla U\Vert _{L^\infty } \Vert u\Vert _{H^{3}}\Vert \partial ^{\alpha } u\Vert _{L^{2}}\\&\le C\Vert \nabla U\Vert _{L^\infty }\Vert u\Vert ^2_{H^{3}}, \end{aligned} \end{aligned}$$
(A.85)
where we use the following inequality:
$$\begin{aligned} \sum _{|\alpha |\le m}\mathop {\int }\limits _{{\mathbb {R}}^3} \Vert \nabla ^{\alpha }( fg)-(\nabla ^{\alpha }f)g\Vert _{L^2}\le C(\Vert f\Vert _{H^{m-1}}\Vert \nabla +\Vert f\Vert _{L^{\infty }}\Vert g\Vert _{H^m}). \end{aligned}$$
For the external force F,
$$\begin{aligned} \begin{aligned} \sum _{0\le |\alpha |\le 2}\mathop {\int }\limits _{{\mathbb {R}}^3} \partial ^{\alpha }F \cdot \partial ^{\alpha }u\,\text {d}x= C\Vert F\Vert ^2_{H^2}+\frac{m_0}{64}\sum _{1\le |\beta |\le 2}\Vert \partial ^{\beta }u\Vert ^2_{L^{2}}. \end{aligned} \end{aligned}$$
(A.86)
We combine (A.71), (A.74), (A.78) and (A.84) with (A.85) and (A.86) to conclude
$$\begin{aligned} \begin{aligned}&\frac{d}{\text {d}t}||u||^2_{H^3({\mathbb {R}}^3)} +\frac{m_0}{2}\mathop {\int }\limits _{{\mathbb {R}}^3} (|\nabla ^3 Du|^2+|\nabla ^2 Du|^2 + |\nabla Du|^2 + | Du|^2)\,\text {d}x\\&\quad \le C\Vert \nabla U\Vert _{L^\infty }\Vert u\Vert ^2_{H^{3}} +C\Vert F\Vert ^2_{H^{2}} +C\Vert G\left[ |Du|^2 \right] \Vert _{L^\infty }(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert _{L^\infty })\Vert \nabla ^2 Du\Vert ^2_{L^2}\\&\qquad +C(\Vert G\left[ |Du|^2 \right] \Vert ^2_{L^\infty }+\Vert G\left[ |Du|^2 \right] \Vert ^6_{L^\infty })(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}. \end{aligned} \end{aligned}$$
(A.87)
Furthermore, we have
$$\begin{aligned} \Vert G\left[ |Du|^2 \right] \Vert _{L^\infty } \le \max _{0\le s\le \Vert Du\Vert _{L^\infty }}G[s] \le \max _{0\le s\le C\Vert u\Vert _{H^3}}G[s]:=g(\Vert u\Vert _{H^3}), \end{aligned}$$
(A.88)
where \(g:[0,\infty ) \mapsto [0,\infty )\) is a non-decreasing function. We set \(X(t):= \Vert u(t)\Vert _{H^3({\mathbb {R}}^3)}\) and it then follows from (A.87) and (A.88) that
$$\begin{aligned} \frac{d}{\text {d}t} X^2\le f_3(X)X^2+C\Vert F\Vert ^2_{H^{2}({\mathbb {R}}^3)} \end{aligned}$$
for some non-decreasing continuous function \(f_3\), which immediately implies that there exists \(T_3>0\) such that \( \sup _{0\le t\le T_3}X(t) < \infty \).
We note that \(\partial _t{u}\in L^2((0,T);L^2({\mathbb {R}}^3))\). Indeed, we introduce the antiderivative of G, denoted by \({\tilde{G}}\), i.e., \({\tilde{G}}[s]=\mathop {\int }\limits ^s_0 G[\tau ]d\tau \). Multiplying \(\partial _t u\) to (4.56), integrating it by parts and using Hölder and Young’s inequalities, we have
$$\begin{aligned} \frac{1}{2}\mathop {\int }\limits _{{\mathbb {R}}^3}|\partial _t{u}|^2\,\text {d}x +\frac{1}{2}\frac{d}{\text {d}t}\mathop {\int }\limits _{{\mathbb {R}}^3}{\tilde{G}}[|Du|^2]\,\text {d}x\le C\mathop {\int }\limits _{{\mathbb {R}}^3}|{\bar{u}}|^2|\nabla u|^2+C\mathop {\int }\limits _{{\mathbb {R}}^3}|F|^2\,\text {d}x. \end{aligned}$$
(A.89)
Again, integrating estimate (A.89) over the time interval [0, T], we obtain
$$\begin{aligned}{} & {} \mathop {\int }\limits _0^{T}\mathop {\int }\limits _{{\mathbb {R}}^3}|\partial _t u|^2\,\text {d}x\text {d}t+\mathop {\int }\limits _{{\mathbb {R}}^3}{\tilde{G}}[|Du(\cdot , T)|^2]\,\text {d}x \nonumber \\{} & {} \quad \le \mathop {\int }\limits _{{\mathbb {R}}^3}{\tilde{G}}[|Du_0|^2]\,\text {d}x+ C\mathop {\int }\limits _0^{T}\mathop {\int }\limits _{{\mathbb {R}}^3}|{\bar{u}}|^2|\nabla u|^2\,\text {d}x\text {d}t+C\mathop {\int }\limits _0^{T}\mathop {\int }\limits _{{\mathbb {R}}^3}|F|^2\,\text {d}x\text {d}t. \end{aligned}$$
(A.90)
Using Sobolev embedding, the second term in (A.90) is estimated as follows:
$$\begin{aligned} \mathop {\int }\limits _0^{T}\mathop {\int }\limits _{{\mathbb {R}}^3}|{\bar{u}}|^2|\nabla u|^2\,\text {d}x\text {d}t\le & {} \mathop {\int }\limits _0^{T} \Vert u\Vert ^2_{L^{\infty }}\Vert \nabla u\Vert ^2_{L^2}\text {d}t \nonumber \\\le & {} C\sup _{0<\tau \le T}\Vert u(\tau )\Vert ^2_{H^{2}}\mathop {\int }\limits _0^{T} \Vert \nabla u\Vert ^2_{L^{2}}\text {d}t<\infty , \end{aligned}$$
(A.91)
and due to the assumption for the external force F, we also get \( \mathop {\int }\limits _0^{T}\mathop {\int }\limits _{{\mathbb {R}}^3}|F|^2\,\text {d}x\text {d}t<\infty \). Therefore, we obtain \(\partial _t{u} \in L^{2}(0,T; L^2({\mathbb {R}}^3))\).
For the uniqueness of a solution, we let \(u_1\) and \(u_2\) be strong solutions for system (4.56). First of all, we rewrite the equation for \({\tilde{u}}:=u_1-u_2\) and \({\tilde{p}}:=p_1-p_2\).
$$\begin{aligned} {\tilde{u}}_t-\nabla \cdot (G(|Du_1|^{2})-G(|Du_2|^{2})) +(U\cdot \nabla ){\tilde{u}}+\nabla {\tilde{p}}=0, \end{aligned}$$
with \(\nabla \cdot {\tilde{u}}=0\) and \(\nabla \cdot U=0\). Multiplying \({\tilde{u}}\) on the both sides of the equation above and integrating on \({\mathbb {R}}^3\), we obtain
$$\begin{aligned} \begin{aligned}&\frac{d}{\text {d}t}\Vert {\tilde{u}}\Vert ^2_{L^2({\mathbb {R}}^3)}+m_0\Vert \nabla {\tilde{u}}\Vert ^2_{L^2({\mathbb {R}}^3)}\le 0, \end{aligned} \end{aligned}$$
(A.92)
where we use Lemma 2.4 and the divergence-free condition. Applying Gronwall’s inequality to estimate (A.92), we get \({\tilde{u}}(x,0)=0\) in \(L^\infty (0,T;L^2({\mathbb {R}}^3))\cap L^2(0,T;H^{1}({\mathbb {R}}^3))\). Hence this implies the uniqueness of a solution. Finally, we introduce Galerkin approximation procedure of equation (4.56)–(4.57) to make up construction of solution. We omit this part (see refer to [13, Proposition 3.1] for detailed proof). \(\square \)
