#### 期刊菜单

Global Well-Posedness for the 2D Incompressible MHD Equations with Vacuum
DOI: 10.12677/AAM.2023.127322, PDF, HTML, XML, 下载: 135  浏览: 162  科研立项经费支持

Abstract: This paper focuses on the global well-posedness for the 2D incompressible Magnetohydrodynamics (MHD) equations with only bounded nonnegative density. We establish the global solutions by using a new a prior estimate without regularity or positive lower bound for the initial density, or compat-ibility conditions. This result generalizes previous result for the 2D Navier-Stokes equations on the periodic domain.

1. 引言与主要结果

$\left\{\begin{array}{l}{\partial }_{t}\rho +div\left(\rho v\right)=0,\hfill \\ {\partial }_{t}\left(\rho u\right)+div\left(\rho u\otimes u\right)+\nabla P=\mu \Delta u+b\cdot \nabla b,\hfill \\ {\alpha }_{t}b+u\cdot \nabla b=\upsilon \Delta b+b\cdot \nabla u,\hfill \\ divu=divb=0.\hfill \end{array}$ (1)

$\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\int }_{\Omega }\rho {|u|}^{2}\text{d}x+{\int }_{\Omega }{|b|}^{2}\text{d}x\right)+{\int }_{\Omega }{|\nabla u|}^{2}\text{d}x+{\int }_{\Omega }{|\nabla b|}^{2}\text{d}x=0,$ (2)

${\int }_{\Omega }\rho u\left(t,x\right)\text{d}x={\int }_{\Omega }{\rho }_{0}{u}_{0}\left(x\right)\text{d}x,$ (3)

${\int }_{\Omega }\rho \left(t,x\right)\text{d}x={\int }_{\Omega }{\rho }_{0}\left(x\right)\text{d}x.$ (4)

$\underset{x\in \Omega }{\mathrm{inf}}\rho \left(t,x\right)=\underset{x\in \Omega }{\mathrm{inf}}{\rho }_{0}\left(t,x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{x\in \Omega }{\mathrm{sup}}\rho \left(t,x\right)=\underset{x\in \Omega }{\mathrm{sup}}{\rho }_{0}\left(t,x\right).$ (5)

$\left\{\begin{array}{l}0\le {\rho }_{0}\le {\rho }^{\text{*}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}M:={\int }_{\Omega }{\rho }_{0}\text{d}x>0,\\ div{u}_{0}=div{b}_{0}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{0},{b}_{0}\in {H}^{1}\left(\Omega \right).\end{array}$ (6)

$\left\{\begin{array}{l}\rho \in {L}^{\infty }\left({R}^{+};{L}^{\infty }\left(\Omega \right)\right),\rho \in C\left({R}^{+};{L}^{p}\left(\Omega \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}10,\\ \sqrt{\rho }u,b\in C\left({R}^{+};{L}^{2}\left(\Omega \right)\right),\sqrt{\rho }{u}_{t},{b}_{t},{\nabla }^{2}u,{\nabla }^{2}b,\nabla P\in {L}^{2}\left({R}^{+};{L}^{2}\left(\Omega \right)\right),\\ \nabla \left(\sqrt{t}P\right),{\nabla }^{2}\left(\sqrt{t}u\right),{\nabla }^{2}\left(\sqrt{t}b\right)\in {L}^{\infty }\left(0,T;{L}^{r}\left(\Omega \right)\right)\cap {L}^{2}\left(R0,T;{L}^{m}\left(\Omega \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}T>0,\\ 1\le r<2,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le m<\infty .\end{array}$ (7)

2. 先验估计

2.1. 符号

${‖f‖}_{q}={‖f‖}_{{L}^{q}\left(\Omega \right)},{H}^{k}={W}^{k,2}.$

2.2. 引理

$\begin{array}{l}{‖f‖}_{{L}^{p}\left({R}^{2}\right)}^{p}\le C{‖f‖}_{{L}^{2}\left({R}^{2}\right)}^{2}{‖\nabla f‖}_{{L}^{2}\left({R}^{2}\right)}^{p-2},\\ {‖g‖}_{C\left(\stackrel{¯}{{R}^{2}}\right)}\le C{‖g‖}_{{L}^{q}\left({R}^{2}\right)}^{q\left(r-2\right)/\left(2r+q\left(r-2\right)\right)}{‖\nabla g‖}_{{L}^{r}\left({R}^{2}\right)}^{2r/\left(2r+q\left(r-2\right)\right)}\end{array}$ (8)

${H}^{1}\left({R}^{2}\right),\text{BMO}\left({R}^{2}\right)$ 代表一般的Hardy于BMO空间，更多的详细可参考文献 [11] 。以下引理在下

${‖u\cdot v‖}_{{H}^{1}\left({R}^{2}\right)}\le C{‖u‖}_{{L}^{2}\left({R}^{2}\right)}{‖v‖}_{{L}^{2}\left({R}^{2}\right)},$ (9)

2) 存在常数 $C>0$ 使得

${‖u‖}_{\text{BMO}\left({R}^{2}\right)}\le C{‖\nabla u‖}_{{L}^{2}\left({R}^{2}\right)},u\in \stackrel{˙}{H}\left({R}^{2}\right)$ (10)

${\left({\int }_{{T}^{2}}\rho {|u|}^{4}\text{d}x\right)}^{\frac{1}{2}}\le C{‖\sqrt{\rho }u‖}_{2}{‖\nabla u‖}_{2}{\mathrm{log}}^{\frac{1}{2}}\left(e+\frac{{‖\rho -M‖}_{2}^{2}}{{M}^{2}}+\frac{{\rho }^{*}{‖\nabla u‖}_{2}^{2}}{{‖\sqrt{\rho }u‖}_{2}^{2}}\right)+C{‖\sqrt{\rho }u‖}_{2}\frac{|{\int }_{{T}^{2}}{\rho }_{0}{u}_{0}\text{d}x|}{M}$ (11)

3. 定理1.1存在性的证明

3.1. Sobolev正则性的持久性

$\underset{t\in \left[0,T\right)}{\mathrm{sup}}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)+\underset{0}{\overset{t}{\int }}\left({‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+{‖{\nabla }^{2}u‖}_{2}^{2}+{‖{\nabla }^{2}b‖}_{2}^{2}+{‖\nabla P‖}_{2}^{2}\right)\text{d}\tau \le C,$ (12)

$\underset{t\in \left[0,T\right)}{\mathrm{sup}}\left({‖\sqrt{\rho }u‖}_{2}^{2}+{‖b‖}_{2}^{2}\right)+2\underset{0}{\overset{T}{\int }}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)\text{d}t\le 2\left({‖\sqrt{{\rho }_{0}}{u}_{0}‖}_{2}^{2}+{‖{b}_{0}‖}_{2}^{2}\right).$ (13)

$\begin{array}{c}{\int }_{{T}^{2}}\rho {|\stackrel{˙}{u}|}^{2}\text{d}x=\underset{{T}^{2}}{\int }\Delta u\cdot \stackrel{˙}{u}\text{d}x-\underset{{T}^{2}}{\int }\nabla P\cdot \stackrel{˙}{u}\text{d}x+\underset{{T}^{2}}{\int }b\cdot \nabla b\cdot \stackrel{˙}{u}\text{d}x\\ ={I}_{1}+{I}_{2}+{I}_{3}.\end{array}$ (14)

$\begin{array}{c}{I}_{1}=\underset{{T}^{2}}{\int }\Delta u\cdot \left({u}_{t}+u\cdot \nabla u\right)\text{d}x\\ =-\frac{1}{2}\underset{{T}^{2}}{\int }\frac{\text{d}}{\text{d}t}{|\nabla u|}^{2}\text{d}x-\underset{{T}^{2}}{\int }{\partial }_{i}{u}^{j}{\partial }_{i}\left({u}^{k}{\partial }_{k}{u}^{j}\right)\text{d}x\\ \le -\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\nabla u‖}_{2}^{2}+C{‖\nabla u‖}_{3}^{3}\\ \le -\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\nabla u‖}_{2}^{2}+C{‖\nabla u‖}_{2}^{2}{‖{\nabla }^{2}u‖}_{2}.\end{array}$ (15)

$\begin{array}{c}{I}_{2}=-\underset{{T}^{2}}{\int }\nabla P\left({u}_{t}+u\cdot \nabla u\right)\text{d}x\\ =\underset{{T}^{2}}{\int }P{\partial }_{j}{u}^{i}{\partial }_{i}{u}^{j}\text{d}x\\ \le C{‖P‖}_{\text{BMO}}{‖{\partial }_{j}{u}^{i}{\partial }_{i}{u}^{j}‖}_{{\text{H}}^{1}}\\ \le C{‖\nabla P‖}_{2}{‖\nabla u‖}_{2}^{2},\end{array}$ (16)

$\begin{array}{c}{I}_{3}=\underset{{T}^{2}}{\int }b\cdot \nabla b\cdot {u}_{t}\text{d}x+{\int }_{{T}^{2}}b\cdot \nabla b\cdot \left(u\cdot \nabla u\right)\text{d}x\\ =-\frac{\text{d}}{\text{d}t}\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot b\text{d}x+\underset{{T}^{2}}{\int }{b}_{t}\cdot \nabla u\cdot b\text{d}x+\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot {b}_{t}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{{T}^{2}}{\int }{b}^{i}{\partial }_{i}{u}^{j}{\partial }_{j}{u}^{k}{b}^{k}\text{d}x-\underset{{T}^{2}}{\int }{b}^{i}{u}^{j}{\partial }_{i}{\partial }_{j}{u}^{k}{b}^{k}\text{d}x\end{array}$

$\begin{array}{c}=-\frac{\text{d}}{\text{d}t}\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot b\text{d}x+\underset{{T}^{2}}{\int }\left(\Delta b-u\cdot \nabla b+b\cdot \nabla u\right)\cdot \nabla u\cdot b\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{{T}^{2}}{\int }b\cdot \nabla u\left(\Delta b-u\cdot \nabla b+b\cdot \nabla u\right)\text{d}x-\underset{{T}^{2}}{\int }{b}^{i}{\partial }_{i}{u}^{j}{\partial }_{j}{u}^{k}{b}^{k}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{{T}^{2}}{\int }{u}^{j}{\partial }_{j}{b}^{i}{\partial }_{i}{u}^{k}{b}^{k}\text{d}x+\underset{{T}^{2}}{\int }{b}^{i}{\partial }_{i}{u}^{k}{u}^{j}{\partial }_{j}{b}^{k}\text{d}x\\ \le -\frac{\text{d}}{\text{d}t}\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot b\text{d}x+\frac{1}{2}{‖{\nabla }^{2}b‖}_{2}^{2}+C{‖b‖}_{2}^{2}{‖\nabla b‖}_{2}^{4}+C{‖\nabla u‖}_{2}^{2}{‖{\nabla }^{2}u‖}_{2}\end{array}$ (17)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\frac{1}{2}{‖\nabla u‖}_{2}^{2}+\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot b\text{d}x\right)+{‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}\\ \le \frac{1}{4}{‖\Delta b‖}_{2}^{2}+C{‖b‖}_{2}^{2}{‖\nabla b‖}_{2}^{4}+C\left({‖{\nabla }^{2}u‖}_{2}+{‖\nabla P‖}_{2}\right){‖\nabla u‖}_{2}^{2}.\end{array}$ (18)

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\nabla b‖}_{2}^{2}+{‖\Delta b‖}_{2}^{2}\le C\underset{{T}^{2}}{\int }|\nabla u|{|\nabla b|}^{2}\text{d}x+C\underset{{T}^{2}}{\int }|\nabla u||b||\Delta b|\text{d}x\\ \le C{‖\nabla u‖}_{3}{‖\nabla b‖}_{2}^{\frac{4}{3}}{‖{\nabla }^{2}b‖}_{2}^{\frac{2}{3}}+C{‖\nabla u‖}_{3}{‖b‖}_{6}{‖\Delta b‖}_{2}\\ \le C{‖\nabla u‖}_{2}^{2}{‖{\nabla }^{2}u‖}_{2}+C\left(1+{‖b‖}_{2}^{2}\right){‖\nabla b‖}_{2}^{4}+\frac{1}{4}{‖\Delta b‖}_{{}_{2}}^{2},\end{array}$ (19)

$\begin{array}{c}-\underset{{T}^{2}}{\int }u\cdot \nabla b\cdot \Delta b\text{d}x=-\underset{{T}^{2}}{\int }{u}^{i}\cdot {\partial }_{i}{b}^{j}{\partial }_{k}^{2}{b}^{j}\text{d}x\\ =\underset{{T}^{2}}{\int }{\partial }_{k}{u}^{i}\cdot {\partial }_{i}{b}^{j}{\partial }_{k}{b}^{j}\text{d}x+\underset{{T}^{2}}{\int }{u}^{i}\cdot {\partial }_{k}{\partial }_{i}{b}^{j}{\partial }_{k}{b}^{j}\text{d}x\\ =\underset{{T}^{2}}{\int }{\partial }_{k}{u}^{i}\cdot {\partial }_{i}{b}^{j}{\partial }_{k}{b}^{j}\text{d}x.\end{array}$

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}+\underset{{T}^{2}}{\int }b\cdot \nabla u\cdot b\text{d}x\right)+{‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+{‖{\nabla }^{2}b‖}_{2}^{2}\\ \le \frac{1}{4}{‖\Delta b‖}_{2}^{4}+C\left({\rho }^{*}\right)\left({‖{\nabla }^{2}u‖}_{2}+{‖\nabla P‖}_{2}\right){‖\nabla u‖}_{2}^{2}\\ \le \frac{1}{4}{‖\Delta b‖}_{2}^{4}+\epsilon {\left({‖{\nabla }^{2}u‖}_{2}+{‖\nabla P‖}_{2}\right)}^{2}+C{‖\nabla u‖}_{2}^{4}.\end{array}$ (20)

${‖\Delta u‖}_{2}^{2}\le \epsilon {‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+\epsilon {‖b‖}_{2}^{2}{‖\nabla b‖}_{2}^{4}+\frac{1}{8}{‖\nabla b‖}_{2}^{4}+\frac{1}{8}{‖{\nabla }^{2}b‖}_{2}^{2},$ (21)

${‖\nabla P‖}_{2}^{2}\le \epsilon {‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+\epsilon {‖b‖}_{2}^{2}{‖\nabla b‖}_{2}^{4}+\frac{1}{8}{‖\nabla b‖}_{2}^{4}+\frac{1}{8}{‖{\nabla }^{2}b‖}_{2}^{2}.$ (22)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}G\left(t\right)+{‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+\left({‖{\nabla }^{2}u‖}_{2}^{2}+{‖{\nabla }^{2}b‖}_{2}^{2}+{‖\nabla P‖}_{2}^{2}\right)\\ \le C{‖\nabla u‖}_{2}^{4}+{‖\nabla b‖}_{2}^{4}+\epsilon {‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+\epsilon {‖b‖}_{2}^{2}{‖\nabla b‖}_{2}^{4},\end{array}$ (23)

$\frac{C}{2}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)\le G\left(t\right)\le C\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right).$ (24)

$\begin{array}{l}G\left(t\right)+{\int }_{{T}^{2}}\left({‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+{‖{\nabla }^{2}u‖}_{2}^{2}+{‖{\nabla }^{2}b‖}_{2}^{2}+{‖\nabla P‖}_{2}^{2}\right)\text{d}t\\ \le C\left({‖\nabla {u}_{0}‖}_{2}^{2}+{‖\nabla {b}_{0}‖}_{2}^{2}\right){\text{e}}^{{\int }_{0}^{T}G\left(t\right)\text{d}t}\\ \le C\left({‖\nabla {u}_{0}‖}_{2}^{2}+{‖\nabla {b}_{0}‖}_{2}^{2}\right)\mathrm{exp}\left(C{\int }_{0}^{T}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)\text{d}t\right),\end{array}$ (25)

$\underset{t\in \left[0,T\right]}{\mathrm{sup}}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)+{\int }_{0}^{T}\left({‖\sqrt{\rho }\stackrel{˙}{u}‖}_{2}^{2}+{‖{\nabla }^{2}u‖}_{2}^{2}+{‖{\nabla }^{2}b‖}_{2}^{2}+{‖\nabla P‖}_{2}^{2}\right)\text{d}t\le C.$ (26)

${‖u\left(t\right)‖}_{p}\le \frac{1}{M}|{\int }_{{T}^{2}}\left({\rho }_{0}{u}_{0}\right)\left(x\right)\text{d}x|+{C}_{p}\left(1+\frac{{‖M-{\rho }_{0}‖}_{2}}{M}\right){‖\nabla u\left(t\right)‖}_{2}.$ (27)

${‖u\left(t\right)‖}_{p}\le |\stackrel{¯}{u}\left(t\right)|+{‖u\left(t\right)-\stackrel{¯}{u}\left(t\right)‖}_{p}\le |\stackrel{¯}{u}\left(t\right)|+{C}_{p}{‖\nabla u\left(t\right)‖}_{2},$ (28)

$M\stackrel{¯}{u}\left(t\right)={\int }_{{T}^{2}}\left(\rho u\right)\left(t,x\right)\text{d}x+{\int }_{{T}^{2}}\left(M-\rho \left(t,x\right)\right)\left(u\left(t,x\right)-\stackrel{¯}{u}\left(t\right)\right)\text{d}x.$ (29)

$|\stackrel{¯}{u}\left(t\right)|\le \frac{1}{M}|{\int }_{{T}^{2}}\left({\rho }_{0}{u}_{0}\right)\left(x\right)\text{d}x|+\frac{{‖M-{\rho }_{0}‖}_{2}}{M}{‖\nabla u\left(t\right)‖}_{2}.$ (30)

${‖u‖}_{{L}^{1}\left(\left[0,T\right];{L}^{\infty }\right)}^{4}\le C{‖u‖}_{{L}^{\infty }\left(\left[0,T\right];{L}^{2}\right)}^{2}{‖{\nabla }^{2}u‖}_{{L}^{2}\left(\left[0,T\right];{L}^{2}\right)}^{2}\le C.$

$\underset{t\in \left[0,T\right)}{\mathrm{sup}}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)+\underset{0}{\overset{t}{\int }}\left({‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}\right)\text{d}t\le C.$ (31)

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\nabla u‖}_{2}^{2}+{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}=-{\int }_{{T}^{2}}\left(\rho u\cdot \nabla u\right)\cdot {u}_{t}\text{d}x+{\int }_{{T}^{2}}b\cdot \nabla b\cdot {u}_{t}\text{d}x\\ \le \frac{1}{2}{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+C{\int }_{{T}^{2}}\rho {|u\cdot \nabla u|}^{2}\text{d}x-\frac{\text{d}}{\text{d}t}{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x+{‖{b}_{t}‖}_{2}{‖\nabla u‖}_{2}{‖b‖}_{\infty }\\ \le \frac{1}{2}{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+C{‖\sqrt{\rho }{|u|}^{2}‖}_{2}{‖\sqrt{\rho }{|\nabla u|}^{2}‖}_{2}-\frac{\text{d}}{\text{d}t}{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x+\frac{1}{4}{‖{b}_{t}‖}_{2}+C{‖\nabla u‖}_{2}^{2}{‖b‖}_{\infty }^{2}\\ \le \frac{1}{2}{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right){‖\sqrt{\rho }{|u|}^{2}‖}_{2}{‖\nabla u‖}_{2}{‖{\nabla }^{2}u‖}_{2}-\frac{\text{d}}{\text{d}t}{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x+\frac{1}{4}{‖{b}_{t}‖}_{2}+C{‖\nabla u‖}_{2}^{2}{‖b‖}_{\infty }^{2}.\end{array}$

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}_{2}^{2}+{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x\right)+\frac{1}{2}{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}\\ \le \frac{1}{4}{‖{b}_{t}‖}_{2}^{2}+C{‖\nabla u‖}_{2}^{2}{‖b‖}_{\infty }^{2}+C{‖{\nabla }^{2}u‖}_{2}^{2}+C\left({\rho }^{*}\right){‖\sqrt{{\rho }_{0}}{u}_{0}‖}_{2}^{2}\frac{{‖\nabla u‖}_{2}^{2}}{{M}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+C\left({\rho }^{*}\right){‖\sqrt{{\rho }_{0}}{u}_{0}‖}_{2}^{2}{‖\nabla u‖}_{2}^{4}\mathrm{log}\left(e+\frac{{‖{\rho }_{0}-M‖}_{2}^{2}}{{M}^{2}}+{\rho }^{*}\frac{{‖\nabla u‖}_{2}^{2}}{{‖\sqrt{{\rho }_{0}}{u}_{0}‖}_{2}^{2}}\right).\end{array}$ (32)

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\nabla b‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}={\int }_{{T}^{2}}\left(b\cdot \nabla u\cdot {b}_{t}-u\cdot \nabla b\cdot {b}_{t}\right)\text{d}x\\ \le \frac{1}{4}{‖{b}_{t}‖}_{2}^{2}+C{‖u‖}_{\infty }^{2}{‖\nabla b‖}_{2}^{2}+C{‖b‖}_{\infty }^{2}{‖\nabla u‖}_{2}^{2}.\end{array}$ (33)

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}+{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x\right)+{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}\\ \le C\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)\left({‖u‖}_{\infty }^{2}+{‖b‖}_{\infty }^{2}+1+C\mathrm{log}\left(e+{‖\nabla u‖}_{2}^{2}\right){‖\nabla u‖}_{2}^{2}\right)+C{‖{\nabla }^{2}u‖}_{2}^{2},\end{array}$

$\begin{array}{l}{‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}+{\int }_{{T}^{2}}b\cdot \nabla u\cdot b\text{d}x+{\int }_{0}^{T}\left({‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}\right)\text{d}t\\ \le C\underset{t\in \left[0,T\right]}{\mathrm{sup}}\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right)\left({\int }_{0}^{T}\left({‖u‖}_{\infty }^{2}+{‖b‖}_{\infty }^{2}+1\right)\text{d}t+C{\int }_{0}^{T}\mathrm{log}\left(e+{‖\nabla u‖}_{2}^{2}\right){‖\nabla u‖}_{2}^{2}\text{d}t\right)+C{\int }_{0}^{T}{‖{\nabla }^{2}u‖}_{2}^{2}\text{d}t,\end{array}$ (34)

3.2. 时间导数的估计

$\left\{\begin{array}{l}\rho {u}_{tt}+{\rho }_{t}{u}_{t}+{\rho }_{t}u\cdot \nabla u+\rho {u}_{t}\cdot \nabla u+\rho u\cdot \nabla {u}_{t}-\Delta {u}_{t}+\nabla {P}_{t}={b}_{t}\cdot \nabla b+b\cdot \nabla {b}_{t}\\ {b}_{tt}+{u}_{t}\cdot \nabla b+u\cdot \nabla {b}_{t}-\Delta {b}_{t}={b}_{t}\cdot \nabla u+b\cdot \nabla {u}_{t}.\end{array}$ (35)

$\begin{array}{l}\rho {\left(\sqrt{t}{u}_{t}\right)}_{t}-\frac{1}{2\sqrt{t}}\rho {u}_{t}+\sqrt{t}{\rho }_{t}{u}_{t}+\sqrt{t}{\rho }_{t}u\cdot \nabla u+\sqrt{t}\rho {u}_{t}\cdot \nabla u+\sqrt{t}\rho u\cdot \nabla {u}_{t}-\Delta \left(\sqrt{t}{u}_{t}\right)+\nabla \left(\sqrt{t}{P}_{t}\right)\\ =\sqrt{t}{b}_{t}\cdot \nabla b+\sqrt{t}b\cdot \nabla {b}_{t},\end{array}$ (36)

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\int }_{{T}^{2}}\rho t{|{u}_{t}|}^{2}\text{d}x+{\int }_{{T}^{2}}t{|\nabla {u}_{t}|}^{2}\text{d}x\\ =\frac{1}{2}{\int }_{{T}^{2}}\rho {|{u}_{t}|}^{2}\text{d}x-\frac{1}{2}{\int }_{{T}^{2}}t{\rho }_{t}{|{u}_{t}|}^{2}\text{d}x-{\int }_{{T}^{2}}\left(\sqrt{t}{\rho }_{t}u\cdot \nabla u\right)\cdot \sqrt{t}{u}_{t}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\int }_{{T}^{2}}\left(\sqrt{t}\rho {u}_{t}\cdot \nabla u\right)\cdot \sqrt{t}{u}_{t}\text{d}x-{\int }_{{T}^{2}}\left(\sqrt{t}\rho u\cdot \nabla {u}_{t}\right)\cdot \sqrt{t}{u}_{t}\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{T}^{2}}\left(\sqrt{t}{b}_{t}\cdot \nabla b\right)\cdot \sqrt{t}{u}_{t}\text{d}x+{\int }_{{T}^{2}}\left(\sqrt{t}b\cdot \nabla {b}_{t}\right)\cdot \sqrt{t}{u}_{t}\text{d}x\\ \cong \frac{1}{2}{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+I{I}_{1}+I{I}_{2}+I{I}_{3}+I{I}_{4}+I{I}_{5}+I{I}_{6}.\end{array}$ (37)

$\begin{array}{c}I{I}_{1}=\frac{1}{2}{\int }_{{T}^{2}}tdiv\left(\rho u\right){|{u}_{t}|}^{2}\text{d}x\le {\int }_{{T}^{2}}t\rho |u||{u}_{t}||\nabla {u}_{t}|\text{d}x\\ \le C{\left({\int }_{{T}^{2}}\rho t{|{u}_{t}|}^{2}\text{d}x\right)}^{\frac{1}{2}}{\left({\int }_{{T}^{2}}t\rho {|u|}^{2}{|\nabla {u}_{t}|}^{2}\text{d}x\right)}^{\frac{1}{2}}\\ \le C\left({\rho }^{*}\right){‖\sqrt{\rho t}{u}_{t}‖}_{2}{‖u‖}_{\infty }{‖\sqrt{t}\nabla {u}_{t}‖}_{2}\\ \le \frac{1}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right){‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}{‖u‖}_{\infty }^{2},\end{array}$

$\begin{array}{c}I{I}_{2}={\int }_{{T}^{2}}t\rho u\cdot \nabla \left[u\cdot \nabla u\cdot {u}_{t}\right]\text{d}x\\ \le {\int }_{{T}^{2}}t\rho |u|\left(|{u}_{t}|{|\nabla u|}^{2}+|u||{\nabla }^{2}u||{u}_{t}|+|u||\nabla u||\nabla {u}_{t}|\right)\text{d}x\\ \le {‖u‖}_{\infty }{‖\sqrt{\rho t}{u}_{t}‖}_{2}{‖{|\nabla u|}^{2}\sqrt{\rho t}‖}_{2}+{‖u‖}_{\infty }^{2}{‖\sqrt{\rho t}{u}_{t}‖}_{2}{‖{|\nabla u|}^{2}\sqrt{\rho t}‖}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{12}{‖\nabla \sqrt{t}{u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right)T{‖u‖}_{\infty }^{4}{‖\nabla u‖}_{2}^{2}\\ \le C\left({\rho }^{*}\right)T{‖\nabla u‖}_{2}^{4}+{‖u‖}_{\infty }^{2}{‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right)T{‖{\nabla }^{2}u‖}_{2}^{2}+{‖u‖}_{\infty }^{4}{‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{12}{‖\nabla \sqrt{t}{u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right)T{‖u‖}_{\infty }^{4}{‖\nabla u‖}_{2}^{2}.\end{array}$

$\begin{array}{c}I{I}_{3}\le {\int }_{{T}^{2}}\rho t|{u}_{t}||\nabla {u}_{t}||u|\text{d}x\\ \le C\left({\rho }^{*}\right){‖u‖}_{\infty }{‖\sqrt{\rho t}{u}_{t}‖}_{2}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}\\ \le \frac{1}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right){‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}{‖u‖}_{\infty }^{2}.\end{array}$

$I{I}_{4}\le \frac{1}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+C\left({\rho }^{*}\right){‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}{‖u‖}_{\infty }^{2}.$

$\begin{array}{c}I{I}_{5}\le {\int }_{{T}^{2}}\rho \sqrt{t}|{b}_{t}||\nabla \sqrt{t}|{u}_{t}|||b|\text{d}x\\ \le {‖b‖}_{\infty }{‖\sqrt{t}\nabla {u}_{t}‖}_{2}{‖\sqrt{t}{b}_{t}‖}_{2}\\ \le \frac{1}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+C{‖\sqrt{t}{b}_{t}‖}_{2}^{2}{‖b‖}_{\infty }^{2}.\end{array}$

$\begin{array}{c}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+\frac{7}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}\le C{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+C\left({\rho }^{*},T\right)\left({‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}\left({‖u‖}_{\infty }^{2}+{‖u‖}_{\infty }^{4}\right)\\ \text{\hspace{0.17em}}\text{ }+{‖u‖}_{\infty }^{4}{‖\nabla u‖}_{2}^{2}+{‖\nabla u‖}_{4}^{4}+{‖{\nabla }^{2}u‖}_{2}^{2}\right)+C{‖b‖}_{\infty }^{2}{‖\sqrt{t}{b}_{t}‖}_{2}^{2}+I{I}_{6},\end{array}$ (38)

${\left(\sqrt{t}{b}_{t}\right)}_{t}-\frac{1}{2\sqrt{t}}{b}_{t}+\sqrt{t}{u}_{t}\cdot \nabla b+\sqrt{t}u\cdot \nabla {b}_{t}-\Delta \left(\sqrt{t}{b}_{t}\right)=\sqrt{t}{b}_{t}\cdot \nabla u+\sqrt{t}b\cdot \nabla {u}_{t}.$

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{‖\sqrt{t}{b}_{t}‖}_{2}^{2}+{‖\sqrt{t}\nabla {b}_{t}‖}_{2}^{2}+\sqrt{t}u\cdot \nabla {b}_{t}-\Delta \left(\sqrt{t}{b}_{t}\right)=\sqrt{t}{b}_{t}\cdot \nabla u+\sqrt{t}b\cdot \nabla {u}_{t}\\ =\frac{1}{2}{‖{b}_{t}‖}_{2}^{2}+{\int }_{{T}^{2}}\left(\sqrt{t}{b}_{t}\cdot \nabla u\cdot \sqrt{t}{b}_{t}-\sqrt{t}{u}_{t}\cdot \nabla b\cdot \sqrt{t}{b}_{t}\right)\text{d}x+{\int }_{{T}^{2}}\sqrt{t}b\cdot \nabla {u}_{t}\cdot \sqrt{t}{b}_{t}\text{d}x\\ \cong \frac{1}{2}{‖{b}_{t}‖}_{2}^{2}+I{I}_{7}+I{I}_{8}.\end{array}$ (39)

$\begin{array}{c}I{I}_{7}\le {‖\nabla u‖}_{2}{‖\sqrt{t}{b}_{t}‖}_{4}^{2}+{‖\nabla b‖}_{2}{‖\sqrt{t}{b}_{t}‖}_{3}{‖\sqrt{t}{u}_{t}‖}_{6}\\ \le C{‖\nabla u‖}_{2}{‖\sqrt{t}{b}_{t}‖}_{2}{‖\sqrt{t}\nabla {b}_{t}‖}_{2}+C{‖\nabla b‖}_{2}{‖\sqrt{t}{b}_{t}‖}_{2}^{\frac{2}{3}}{‖\sqrt{t}\nabla {b}_{t}‖}_{2}^{\frac{1}{3}}{‖\sqrt{t}{b}_{t}‖}_{2}^{\frac{1}{3}}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{\frac{2}{3}}\\ \le \frac{1}{12}{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+\frac{1}{2}{‖\sqrt{t}\nabla {b}_{t}‖}_{2}^{2}+C\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}\right){‖\sqrt{t}{b}_{t}‖}_{2}^{2}+C{‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}.\end{array}$ (40)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+{‖\sqrt{t}{b}_{t}‖}_{2}^{2}\right)+{‖\sqrt{t}\nabla {u}_{t}‖}_{2}^{2}+{‖\sqrt{t}\nabla {b}_{t}‖}_{2}^{2}\\ \le C\left({\rho }^{*},T\right)\left({‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}\left(1+{‖u‖}_{\infty }^{2}+{‖u‖}_{\infty }^{4}\right)+{‖u‖}_{\infty }^{4}{‖\nabla u‖}_{2}^{2}+{‖\nabla u‖}_{4}^{4}+{‖{\nabla }^{2}u‖}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}+C{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}+C\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}+{‖b‖}_{\infty }^{2}\right){‖\sqrt{t}{b}_{t}‖}_{2}^{2}.\end{array}$

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+{‖\sqrt{t}{b}_{t}‖}_{2}^{2}+{\int }_{0}^{t}\tau \left({‖\nabla {u}_{t}‖}_{2}^{2}+{‖\nabla {b}_{t}‖}_{2}^{2}\right)\text{d}\tau \right)\\ \le h\left(t\right)\left(1+{‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+{‖\sqrt{t}{b}_{t}‖}_{2}^{2}\right),\end{array}$ (41)

$\begin{array}{l}h\left(t\right):=C\left({\rho }^{*},T\right)\left(1+{‖u‖}_{\infty }^{2}+{‖u‖}_{\infty }^{4}+{‖u‖}_{\infty }^{4}{‖\nabla u‖}_{2}^{2}+{‖\nabla u‖}_{2}^{4}+{‖{\nabla }^{2}u‖}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{‖\sqrt{\rho }{u}_{t}‖}_{2}^{2}+{‖{b}_{t}‖}_{2}^{2}+C\left({‖\nabla u‖}_{2}^{2}+{‖\nabla b‖}_{2}^{2}+{‖b‖}_{\infty }^{2}\right)\end{array}$

$\underset{t\to {0}^{+}}{\mathrm{lim}}\underset{{T}^{2}}{\int }\left(\rho t{|{u}_{t}\left(t,x\right)|}^{2}+t{|{b}_{t}\left(t,x\right)|}^{2}\right)\text{d}x=0.$

${‖\sqrt{\rho t}{u}_{t}‖}_{2}^{2}+{‖\sqrt{t}{b}_{t}‖}_{2}^{2}+{\int }_{0}^{t}\tau \left({‖\nabla {u}_{t}‖}_{2}^{2}+{‖\nabla {b}_{t}‖}_{2}^{2}\right)\text{d}\tau \le \text{exp}\left\{{\int }_{0}^{t}h\left(\tau \right)\text{d}\tau \right\}-1.$ (42)

$\underset{t\in \left[{t}_{0},{t}_{0}+T\right]}{\mathrm{sup}}\underset{{T}^{2}}{\int }\left(\rho \left(t-{t}_{0}\right){|{u}_{t}|}^{2}+\left(t-{t}_{0}\right){|{b}_{t}|}^{2}\right)\text{d}x+{\int }_{{t}_{0}}^{{t}_{0}+T}\underset{{T}^{2}}{\int }\left(t-{t}_{0}\right)\left({|\nabla {u}_{t}|}^{2}+{|\nabla {b}_{t}|}^{2}\right)\text{d}x\text{ }\text{d}t\le C\left(T\right),$ (43)

${‖\sqrt{t}{u}_{t}‖}_{{L}^{2}\left(\left[0,T\right];{L}^{p}\right)}+{‖\sqrt{t}{b}_{t}‖}_{{L}^{2}\left(\left[0,T\right];{L}^{p}\right)}\le C\left(T\right).$ (44)

${\stackrel{¯}{u}}_{t}$${u}_{t}$ 在T2上的平均，可直接计算出

$\underset{{T}^{2}}{\int }\rho {u}_{t}\text{d}x=M{\stackrel{¯}{u}}_{t}+\underset{{T}^{2}}{\int }\rho \left({u}_{t}-{\stackrel{¯}{u}}_{t}\right)\text{d}x,$ (45)

$M|{\stackrel{¯}{u}}_{t}|\le {‖\rho ‖}_{2}{‖\nabla {u}_{t}‖}_{2}+{M}^{\frac{1}{2}}{‖\sqrt{\rho }{u}_{t}‖}_{2}.$

${‖{u}_{t}‖}_{p}\le {‖{u}_{t}-{\stackrel{¯}{u}}_{t}‖}_{p}+|{\stackrel{¯}{u}}_{t}|\le \left({C}_{p}+\frac{{‖{\rho }_{0}‖}_{2}}{M}\right){‖\nabla {u}_{t}‖}_{2}+\frac{1}{{M}^{1/2}}{‖\sqrt{\rho }{u}_{t}‖}_{2}.$ (46)

$N{\stackrel{¯}{b}}_{t}=\underset{{T}^{2}}{\int }a{u}_{t}\text{d}x+\underset{{T}^{2}}{\int }\left(N-a\right)\left({b}_{t}-{\stackrel{¯}{b}}_{t}\right)\text{d}x,$ (47)

${‖{b}_{t}‖}_{p}\le \frac{{‖{\rho }_{0}‖}_{2}}{M}{‖{b}_{t}‖}_{2}+\left(\frac{{‖M-{\rho }_{0}‖}_{2}}{M}+{C}_{p}\right){‖\nabla {b}_{t}‖}_{2}.$ (48)

3.3. 可积性的转移

${‖{\nabla }^{2}\sqrt{t}u‖}_{{L}^{p}\left(0,T;{L}^{{p}^{\ast }-\epsilon }\right)}+{‖{\nabla }^{2}\sqrt{t}b‖}_{{L}^{p}\left(0,T;{L}^{{p}^{\ast }-\epsilon }\right)}+{‖\nabla \sqrt{t}P‖}_{{L}^{p}\left(0,T;{L}^{{p}^{\ast }-\epsilon }\right)}\le {C}_{0,T},$ (49)

${\int }_{0}^{T}{‖\nabla u\left(r\right)‖}_{{L}^{\infty }}^{s}\text{d}t+{\int }_{0}^{T}{‖\nabla b\left(t\right)‖}_{{L}^{\infty }}^{s}\text{d}t\le {C}_{0,T}{T}^{\beta }.$ (50)

$\left\{\begin{array}{l}-\Delta \sqrt{t}u+\nabla \sqrt{t}P=-\rho \sqrt{t}{u}_{t}-\sqrt{t}\rho u\cdot \nabla u+\sqrt{t}b\cdot \nabla b,\\ +u\cdot \nabla {b}_{t}-\Delta \sqrt{t}b=-\sqrt{t}{b}_{t}-\sqrt{t}u\cdot \nabla b+\sqrt{t}b\cdot \nabla u,\\ div\sqrt{t}u=div\sqrt{t}b=0.\end{array}$ (51)

${‖\rho \sqrt{t}{u}_{t}‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}+{‖\sqrt{t}{b}_{t}‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}\le {C}_{0,T},$

${‖\nabla u‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}\le {‖\nabla u‖}_{{L}^{\infty }\left(0,T;{L}^{2}\right)}^{\theta }{‖\nabla u‖}_{{L}^{2}\left(0,T;{H}^{1}\right)}^{1-\theta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta \in \left[0,1\right],$

$\begin{array}{l}{‖\sqrt{t}\rho u\cdot \nabla u‖}_{{L}^{p}\left(0,T;{L}^{r}\right)},{‖\sqrt{t}b\cdot \nabla b‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}\le {C}_{0,T},\\ {‖\sqrt{t}b\cdot \nabla u‖}_{{L}^{p}\left(0,T;{L}^{r}\right)},{‖\sqrt{t}u\cdot \nabla b‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}\le {C}_{0,T}.\end{array}$ (52)

${‖{\nabla }^{2}\sqrt{t}u,{\nabla }^{2}\sqrt{t}b,\nabla \sqrt{t}P‖}_{{L}^{p}\left(0,T;{L}^{r}\right)}\le {C}_{0,T},$ (53)

$\begin{array}{l}{\left({\int }_{0}^{T}{‖\nabla u\left(t\right)‖}_{{L}^{\infty }}^{s}\text{d}t\right)}^{\frac{1}{s}}+{\left({\int }_{0}^{T}{‖\nabla b\left(t\right)‖}_{{L}^{\infty }}^{s}\text{d}t\right)}^{\frac{1}{s}}\\ \le {\left({\int }_{0}^{T}{t}^{-\frac{1}{2}}{‖\nabla \sqrt{t}u\left(t\right)‖}_{{W}^{1,r}}^{s}\text{d}t\right)}^{\frac{1}{s}}+{\left({\int }_{0}^{T}{t}^{-\frac{1}{2}}{‖\nabla \sqrt{t}b\left(t\right)‖}_{{W}^{1,r}}^{s}\text{d}t\right)}^{\frac{1}{s}}\\ \le {\left({\int }_{0}^{T}{t}^{-\frac{ps}{2p-2s}}\text{d}t\right)}^{\frac{1}{s}-\frac{1}{p}}\left({‖\nabla \sqrt{t}u\left(t\right)‖}_{{L}^{p}\left(0,T;{W}^{1,r}\right)}+{‖\nabla \sqrt{t}b\left(t\right)‖}_{{L}^{p}\left(0,T;{W}^{1,r}\right)}\right)\\ \le {C}_{0,T}{T}^{\frac{2p-2s-ps}{2ps}}.\end{array}$

4. 定理1.1唯一性的证明

$\frac{\text{d}X}{\text{d}t}=u\left(t,x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{X|}_{t=0}=y,y\in \Omega .$

$\begin{array}{l}\eta \left(t,y\right)=\rho \left(t,X\left(t,y\right)\right),\\ \left(v,h\right)\left(t,y\right)=\left(u,b\right)\left(t,X\left(t,y\right)\right),\\ Q\left(t,y\right)=P\left(t,X\left(t,y\right)\right),\end{array}$ (54)

$X\left(t,y\right)=y+{\int }_{0}^{t}u\left(\tau ,X\left(\tau ,y\right)\right)\text{d}\tau ={\int }_{0}^{t}v\left(\tau ,X\left(\tau ,y\right)\right)\text{d}\tau ,$

${\nabla }_{y}X\left(t,y\right)=Id+{\int }_{0}^{t}{\nabla }_{y}v\left(\tau ,y\right)\text{d}\tau .$

${\nabla }_{v}:{=}^{T}A{\nabla }_{y},\text{\hspace{0.17em}}\text{\hspace{0.17em}}di{v}_{v}:{=}^{T}A:{\nabla }_{y}=di{v}_{y}\left(A\cdot \right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Delta }_{v}:=di{v}_{y}\left({A}^{T}A{\nabla }_{y\cdot }\right),$

$\left\{\begin{array}{l}{\eta }_{t}=0,\\ \eta {v}_{t}-{\Delta }_{v}v+{\nabla }_{v}Q=h\cdot {\nabla }_{v}h,\\ {h}_{t}-{\Delta }_{v}h=h\cdot {\nabla }_{v}v,\\ di{v}_{v}v=di{v}_{v}h=0.\end{array}$ (55)

$A={\left(Id+\left({\nabla }_{y}X-Id\right)\right)}^{-1}=\underset{k=0}{\overset{+\infty }{\sum }}{\left(-1\right)}^{k}{\left({\int }_{0}^{t}{\nabla }_{y}v\left(\tau ,\cdot \right)\text{d}\tau \right)}^{k}.$

$\left\{\begin{array}{l}{\rho }_{0}\delta {v}_{t}-{\Delta }_{{v}_{1}}\delta v+{\nabla }_{{v}_{1}}\delta Q=\left({\Delta }_{{v}_{2}}-{\Delta }_{{v}_{1}}\right){v}_{2}-\left({\nabla }_{{v}_{2}}-{\nabla }_{{v}_{1}}\right){Q}_{2}-\delta h\cdot {\nabla }_{{v}_{1}}{h}_{1}-{h}_{2}\cdot \left({\nabla }_{{v}_{2}}-{\nabla }_{{v}_{1}}\right){h}_{1}-{h}_{2}\cdot {\nabla }_{{v}_{2}}\delta h,\\ \delta {h}_{t}-{\Delta }_{{v}_{1}}\delta h=\left({\Delta }_{{v}_{2}}-{\Delta }_{{v}_{1}}\right){h}_{2}-\delta h\cdot {\nabla }_{{v}_{1}}{v}_{1}-{h}_{2}\cdot \left({\nabla }_{{v}_{2}}-{\nabla }_{{v}_{1}}\right){v}_{1}-{h}_{2}\cdot {\nabla }_{{v}_{2}}\delta v\\ di{v}_{{v}_{1}}\delta v=\left(di{v}_{{v}_{1}}-di{v}_{{v}_{2}}\right){v}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}di{v}_{{v}_{1}}\delta h=\left(di{v}_{{v}_{1}}-di{v}_{{v}_{2}}\right){h}_{2}.\end{array}$ (56)

5. 结论

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