# 三维不可压磁流体力学方程弱解正则准则Regularity Criterion of the Weak Solution for the 3D Incompressible MHD Equations

DOI: 10.12677/PM.2021.114077, PDF, HTML, XML, 下载: 20  浏览: 68  科研立项经费支持

Abstract: This paper considers the regularity of weak solutions for incompressible MHD equations in 3D cases. Here, Yuong inequalities, Hölder inequalities and Sobolev embedding techniques are used to expand the integral space to which the weak solution belongs. Here, it is proved that the weak solution (u, b) is regular on (0,T], if ∂3u,∂3b∈Lp(0,T;Lq(R3)) and 2/p+3/q=46/25+3/25q, 31/8≤q≤∞ or ∂3u,∂3b∈Lp(0,T;Lq(R3)), 2/p+3/q=22/13+3/13q, 19/8≤q≤∞ ,together with .

1. 引言及主要结论

$\left\{\begin{array}{l}{\partial }_{t}u+\left(u\cdot \nabla \right)u+\nabla p=\Delta u+\left(b\cdot \nabla \right)b,\hfill \\ {\partial }_{t}b+\left(u\cdot \nabla \right)b=\Delta b+\left(b\cdot \nabla \right)u,\hfill \\ \nabla \cdot u=\nabla \cdot b=0,\hfill \\ u\left(x,0\right)={u}_{0}\left(x\right),b\left(x,0\right)={b}_{0}\left(x\right),\hfill \end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,t\right)\in {R}^{3}×\left(0,T\right]$ (1)

${\partial }_{3}\left(u±b\right)\in {L}^{p}\left(\left(0,T\right);{L}^{q}\left({R}^{3}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{2}{p}+\frac{3}{q}=\frac{8}{5}+\frac{3}{5q},4\le q\le \infty$

$\left\{\begin{array}{l}b,{u}_{3}\in {L}^{{p}_{1}}\left(0,T;{L}^{{q}_{1}}\left({R}^{3}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2/{p}_{1}+3/{q}_{1}\le 1,3<{q}_{1}\le \infty ,\\ \partial b,{\partial }_{3}u\in {L}^{{p}_{2}}\left(0,T;{L}^{{q}_{2}}\left({R}^{3}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2/{p}_{2}+3/{q}_{2}\le 2,3/2<{q}_{2}\le \infty .\end{array}$ (2)

${\partial }_{3}w\in {L}^{p}\left(\left(0,T\right);{L}^{q}\left({R}^{3}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{2}{p}+\frac{3}{q}=\frac{46}{25}+\frac{3}{25q},\frac{31}{8}\le q\le \infty$ (3)

${\partial }_{3}w\in {L}^{p}\left(\left(0,T\right);{L}^{q}\left({R}^{3}\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{2}{p}+\frac{3}{q}=\frac{22}{13}+\frac{3}{13q},\frac{19}{8}\le q\le \infty$ (4)

2. 主要结果的证明

$\left\{\begin{array}{l}{\partial }_{t}{w}^{+}+{w}^{-}\cdot \nabla {w}^{+}+\nabla p=\Delta {w}^{+},\hfill \\ {\partial }_{t}{w}^{-}+{w}^{+}\cdot \nabla {w}^{-}+\nabla p=\Delta {w}^{-},\hfill \\ \nabla \cdot {w}^{+}=\nabla \cdot {w}^{-}=0,\hfill \\ {w}^{+}\left(x,0\right)={w}_{0}^{+}\left(x\right),{w}^{-}\left(x,0\right)={w}_{0}^{-}\left(x\right),\hfill \end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,t\right)\in {R}^{3}×\left(0,T\right]$ (5)

${‖u‖}_{{L}^{3q}}\le C{‖{\partial }_{1}u‖}_{{L}^{2}}^{\frac{1}{3}}{‖{\partial }_{2}u‖}_{{L}^{2}}^{\frac{1}{3}}{‖{\partial }_{3}u‖}_{{L}^{q}}^{\frac{1}{3}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\le q\le \infty$. (6)

i) ${‖P‖}_{{L}^{r}}\le C{‖{w}^{+}‖}_{{L}^{2r}}{‖{w}^{-}‖}_{{L}^{2r}},$ (7)

ii) ${‖{\partial }_{3}P‖}_{{L}^{r}}\le C{‖{w}^{-}\cdot {\partial }_{3}{w}^{+}‖}_{{L}^{r}},$ (8)

iii) ${‖{\partial }_{3}P‖}_{{L}^{r}}\le C{‖{w}^{+}\cdot {\partial }_{3}{w}^{-}‖}_{{L}^{r}}.$ (9)

$\frac{19}{8}\le q\le \infty$ 时， ${L}^{\frac{13q}{11q-18}}\left(0,T;{L}^{q}\left({R}^{3}\right)\right)\subset {L}^{\frac{2q}{2q-3}}\left(0,T;{L}^{q}\left({R}^{3}\right)\right)$

${u}_{3},{b}_{3}\in {L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)$.

$\begin{array}{l}{‖{w}^{+}\left(\cdot ,t\right)‖}_{{L}^{2}}^{2}+{‖{w}^{-}\left(\cdot ,t\right)‖}_{{L}^{2}}^{2}+2{\int }_{0}^{t}\left({‖\nabla {w}^{+}\left(\cdot ,\tau \right)‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{-}\left(\cdot ,\tau \right)‖}_{{L}^{2}}^{2}\right)\text{d}\tau \\ \le {‖{w}^{+}\left(\cdot ,0\right)‖}_{{L}^{2}}^{2}+{‖{w}^{-}\left(\cdot ,0\right)‖}_{{L}^{2}}^{2}\le C\end{array}$. (10)

$|{w}_{3}^{+}|{w}_{3}^{+},|{w}_{3}^{-}|{w}_{3}^{-}$ 分别与方程(5)的第一个和第二个方程作内积，利用 $\nabla \cdot {w}^{+}=\nabla \cdot {w}^{-}=0$，然后相加得：

$\begin{array}{l}\frac{1}{3}\frac{\text{d}}{\text{d}t}\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{3}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{3}\right)+\frac{4}{9}{\int }_{{R}^{3}}\left({|\nabla {|{w}_{3}^{+}|}^{3/2}|}^{2}+{|\nabla {|{w}_{3}^{-}|}^{3/2}|}^{2}\right)\text{d}x\\ =-{\int }_{{R}^{3}}{\partial }_{3}p\cdot {w}_{3}^{+}|{w}_{3}^{+}|\text{d}x-{\int }_{{R}^{3}}{\partial }_{3}p\cdot {w}_{3}^{-}|{w}_{3}^{-}|\text{d}x\\ ={I}_{1}+{I}_{2}\end{array}$. (11)

$|{I}_{1}|\le C{‖{\partial }_{3}p‖}_{{L}^{3}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}$ (由Hölder不等式)

$\le C{‖{w}^{+}\cdot {\partial }_{3}{w}^{-}‖}_{{L}^{3}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}$ (由式(8))

$\begin{array}{l}\le C{‖{w}^{+}‖}_{{L}^{3q/\left(q-3\right)}}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}\\ \le C{‖{w}^{+}‖}_{{L}^{2}}^{\left(8q-31\right)/3\left(4q-3\right)}{‖{w}^{+}‖}_{{L}^{3q}}^{\left(q+18\right)/2\left(4q-3\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}\\ \le C{‖{\partial }_{1}{w}^{+}‖}_{{L}^{2}}^{\left(q+18\right)/6\left(4q-3\right)}{‖{\partial }_{2}{w}^{+}‖}_{{L}^{2}}^{\left(q+18\right)/6\left(4q-3\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{\left(q+18\right)/6\left(4q-3\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}\\ \le C{‖\nabla {w}^{+}‖}_{{L}^{2}}^{\left(q+18\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{\left(q+18\right)/6\left(4q-3\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{\left(q+18\right)/\left(23q-36\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{6\left(4q-3\right)/\left(23q-36\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}\end{array}$ (12)

$\le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}$

$|{I}_{2}|\le C\left({‖\nabla {w}^{-}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}$. (13)

$|{I}_{1}|$$|{I}_{2}|$ 的估计式(12)和(13)代入式(11)得

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{3}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{3}\right)\\ \le \frac{\text{d}}{\text{d}t}\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{3}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{3}\right)+\frac{4}{9}\left({‖\nabla {|{w}_{3}^{+}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}+{‖\nabla {|{w}_{3}^{-}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}\right)\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{-}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}\right)\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{2}\right)\end{array}$. (14)

${‖{w}_{3}^{+}‖}_{{L}^{\infty }\left(0,T;{L}^{3}\left({R}^{3}\right)\right)}+{‖{w}_{3}^{-}‖}_{{L}^{\infty }\left(0,T;{L}^{3}\left({R}^{3}\right)\right)}\le C$.

$\begin{array}{l}\frac{4}{9}\left({‖\nabla {|{w}_{3}^{+}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}+{‖\nabla {|{w}_{3}^{-}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}\right)\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{-}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{25q/\left(23q-36\right)}\right)\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{2}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{2}\right)\le C\end{array}$.

$\begin{array}{c}{‖{w}_{3}^{+}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}+{‖{w}_{3}^{-}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}={‖{|{w}_{3}^{+}|}^{\frac{3}{2}}‖}_{{L}^{2}\left(0,T;{L}^{6}\left({R}^{3}\right)\right)}^{\frac{2}{3}}+{‖{|{w}_{3}^{-}|}^{\frac{3}{2}}‖}_{{L}^{2}\left(0,T;{L}^{6}\left({R}^{3}\right)\right)}^{\frac{2}{3}}\\ \le {‖\nabla {|{w}_{3}^{+}|}^{\frac{3}{2}}‖}_{{L}^{2}\left(0,T;{L}^{2}\left({R}^{3}\right)\right)}^{\frac{2}{3}}+{‖\nabla {|{w}_{3}^{-}|}^{\frac{3}{2}}‖}_{{L}^{2}\left(0,T;{L}^{2}\left({R}^{3}\right)\right)}^{\frac{2}{3}}\le C\end{array}$.

$\begin{array}{c}{‖{u}_{3}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}=\frac{1}{2}{‖\left({u}_{3}+{b}_{3}\right)+\left({u}_{3}-{b}_{3}\right)‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}\\ \le \frac{1}{2}\left({‖{w}_{3}^{+}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}+{‖{w}_{3}^{-}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}\right)\le C\end{array}$.

${‖{b}_{3}‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}=\frac{1}{2}{‖\left({u}_{3}+{b}_{3}\right)-\left({u}_{3}-{b}_{3}\right)‖}_{{L}^{3}\left(0,T;{L}^{9}\left({R}^{3}\right)\right)}\le C$.

$\begin{array}{c}|{I}_{1}|=|{\int }_{{R}^{3}}{\partial }_{3}p\cdot {w}_{3}^{+}|{w}_{3}^{+}|\text{d}x|\\ \le C{\int }_{{R}^{3}}|p||{\partial }_{3}{w}_{3}^{+}||{w}_{3}^{+}|\text{d}x\\ \le C{‖p‖}_{{L}^{3q/\left(2q-3\right)}}{‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C\left({‖{w}^{+}‖}_{{L}^{6q/\left(2q-3\right)}}^{2}+{‖{w}^{-}‖}_{{L}^{6q/\left(2q-3\right)}}^{2}\right){‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C{‖{w}^{+}‖}_{{L}^{2}}^{2\left(8q-19\right)/3\left(4q-3\right)}{‖{w}^{+}‖}_{{L}^{3q}}^{2\left(q+9\right)/2\left(4q-3\right)}{‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+C{‖{w}^{-}‖}_{{L}^{2}}^{2\left(8q-19\right)/3\left(4q-3\right)}{‖{w}^{-}‖}_{{L}^{3q}}^{2\left(q+9\right)/2\left(4q-3\right)}{‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C{‖{\partial }_{1}{w}^{+}‖}_{{L}^{2}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{2}{w}^{+}‖}_{{L}^{2}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+C{‖{\partial }_{1}{w}^{-}‖}_{{L}^{2}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{2}{w}^{-}‖}_{{L}^{2}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}_{3}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C{‖\nabla {w}^{+}‖}_{{L}^{2}}^{2\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{13q/3\left(4q-3\right)}{‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+C{‖\nabla {w}^{-}‖}_{{L}^{2}}^{2\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{\left(q+9\right)/3\left(4q-3\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}{‖{w}_{3}^{+}‖}_{{L}^{3}}\end{array}$ (15)

$\begin{array}{l}\le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+C\left({‖\nabla {w}^{-}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{\left(q+9\right)/\left(11q-18\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{3\left(4q-3\right)/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}_{}^{+}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+C\left({‖\nabla {w}^{-}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}_{}^{-}‖}_{{L}^{q}}^{\left(q+9\right)/\left(11q-18\right)}{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{3\left(4q-3\right)/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}\end{array}$

$|{I}_{2}|\le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}\right){‖{w}_{3}^{+}‖}_{{L}^{3}}$. (16)

$|{I}_{1}|$$|{I}_{2}|$ 的估计式(15)和(16)代入式(11)得

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{3}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{3}\right)\\ \le \frac{\text{d}}{\text{d}t}\left({‖{w}_{3}^{+}‖}_{{L}^{3}}^{3}+{‖{w}_{3}^{-}‖}_{{L}^{3}}^{3}\right)+\frac{4}{9}\left({‖\nabla {|{w}_{3}^{+}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}+{‖\nabla {|{w}_{3}^{-}|}^{\frac{3}{2}}‖}_{{L}^{2}}^{2}\right)\\ \le C\left({‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖\nabla {w}^{+}‖}_{{L}^{2}}^{2}+{‖{\partial }_{3}{w}^{+}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}+{‖{\partial }_{3}{w}^{-}‖}_{{L}^{q}}^{13q/\left(11q-18\right)}\right)\left({‖{w}_{3}^{+}‖}_{{L}^{3}}+{‖{w}_{3}^{-}‖}_{{L}^{3}}\right)\end{array}$

${‖{w}_{3}^{+}‖}_{{L}^{\infty }\left(0,T;{L}^{3}\left({R}^{3}\right)\right)}+{‖{w}_{3}^{-}‖}_{{L}^{\infty }\left(0,T;{L}^{3}\left({R}^{3}\right)\right)}\le C$.

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