## Study on Weak Solution and Strong Solution of Incompressible MHD Equations with Damping in Three-Dimensional Systems

Li Kai,, Yang Han, Wang Fan

 基金资助: 国家自然科学基金.  11701477

 Fund supported: the NSFC.  11701477

Abstract

In this paper, the Cauchy problem of the MHD equations with damping is studied. When $\beta \ge 1$ and initial data satisfy ${u_0}$, ${b_0} \in {L^2}({{\mathbb{R} ^3}})$, the Galerkin method is used to prove the global weak solution of the equations. When the initial data satisfy ${u_0} \in H_0^1 \cap {L^{\beta + 1}}({{\mathbb{R} ^3}})$, ${b_0} \in H_0^1({{\mathbb{R} ^3}})$, it is possible to obtain a unique local strong solution for the equation group.

Keywords： MHD equations ; Damping ; Weak solutions ; Strong solutions

Li Kai, Yang Han, Wang Fan. Study on Weak Solution and Strong Solution of Incompressible MHD Equations with Damping in Three-Dimensional Systems. Acta Mathematica Scientia[J], 2019, 39(3): 518-528 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} {u_t} + u \cdot \nabla u + \nabla p - \mu \Delta u + \eta |u{|^{\beta - 1}}u = b \cdot \nabla b, \\ {b_t} + u \cdot \nabla b - \nu \Delta b = b \cdot \nabla u, \\ \nabla \cdot u = 0, \nabla \cdot b = 0, \\ u{|_{t = 0}} = {u_0}, b{|_{t = 0}} = {b_0}, \\ |u| \to 0, |b| \to 0, \quad {\rm as}\ |x| \to 0. \end{array}\right. \end{eqnarray}$

$\begin{eqnarray} && - \int_0^T {(u, {\phi _t})} {\rm d}t + \int_0^T {\int_{{{\Bbb R} ^3}} {(u \cdot \nabla )u\phi } } {\rm d}x{\rm d}t + \int_0^T {\int_{{{\Bbb R} ^3}} {\nabla u:\nabla \phi } } {\rm d}x{\rm d}t \\ &&+ \int_0^T {\int_{{{\Bbb R} ^3}} {|u{|^{\beta - 1}}u\phi } } {\rm d}x{\rm d}t - \int_0^T {\int_{{{\Bbb R} ^3}} b } \nabla b\phi {\rm d}x{\rm d}t = ({u_0}, \phi (0)), \\ & &- \int_0^T {(b, {\phi _t})} {\rm d}t + \int_0^T {\int_{{{\Bbb R} ^3}} {(u \cdot \nabla )b\phi } } {\rm d}x{\rm d}t + \int_0^T {\int_{{{\Bbb R} ^3}} {\nabla b:\nabla \phi } } {\rm d}x{\rm d}t \\ && - \int_0^T {\int_{{{\Bbb R} ^3}} b } \nabla u\phi {\rm d}x{\rm d}t = ({b_0}, \phi (0)). \end{eqnarray}$

(3) $\nabla \cdot u(x, t) $$\nabla \cdot b(x, t)$$ {{\Bbb R} ^3} \times \left[{0, T} \right]$上几乎处处为$0$.

$$$\begin{array}{l} u \in {L^\infty }(0 , T ;{L^2}({{\Bbb R} ^3} )) \cap {L^2} (0 , T ;H_0^1({{\Bbb R} ^3})) \cap {L^{\beta + 1}}(0 , T ;{L^{\beta + 1}}({{\Bbb R} ^3})), \\ b \in {L^\infty }(0 , T ;{L^2}({{\Bbb R} ^3} )) \cap {L^2} (0 , T ;H_0^1({{\Bbb R} ^3})), \end{array}$$$

$\begin{eqnarray} && \mathop {\sup }\limits_{0 \le t \le T} ||u||_{{L^2}}^2 + \mathop {\sup }\limits_{0 \le t \le T} ||b||_{{L^2}}^2 + 2\int_0^T {||\nabla u||_{{L^2}}^2} {\rm d}t + 2\int_0^T {||\nabla b||_{{L^2}}^2} {\rm d}t\\ && + 2\int_0^T {||u||_{{L^{\beta + 1}}}^{\beta + 1}} {\rm d}t \le ||{u_0}||_{{L^2}}^2 + ||{b_0}||_{{L^2}}^2. \end{eqnarray}$

因为$H_0^1$是可分空间且$C_0^\infty $$H_0^1 中稠密.因此存在着 C_0^\infty 中的序列 {w_1}, {w_2}, \cdots, {w_m} ,对于任意的 m ,分别定义逼近解 {u_m} , {b_m} 如下 并且 \begin{eqnarray} &&({u_m}'(t), {w_j}) + ({u_m}(t) \cdot \nabla {u_m}(t), {w_j}) + (\nabla {u_m}(t), \nabla {w_j}) + (|{u_m}{|^{\beta - 1}}{u_m}(t), {w_j})\\ && = ({b_m}(t) \cdot \nabla {b_m}(t), {w_j}), \end{eqnarray} \begin{eqnarray} && ({b_m}'(t), {w_j}) + ({u_m}(t) \cdot \nabla {b_m}(t), {w_j}) + (\nabla {b_m}(t), \nabla {w_j}) = ({b_m}(t) \cdot \nabla {u_m}(t), {w_j}), \end{eqnarray} t \in \left[{0, T} \right] , j = 1, 2, 3, \cdots, m ,并且在 {L^2}$$ {u_{m0}} \to {u_0}$, ${b_{m0}} \to {b_0} $$m \to \infty . 这里需要先得出一个对于 {u_m}, {b_m} 的先验估计. 引理2.1 假设 {u_m}, {b_m} \in {L^2} ,那么对于任意给定的 T > 0 ,任意的 \beta \ge 1 ,有 将(2.4)式两端同时乘以 {g_{jm}}(t) ,然后关于 j = 1, 2, \cdots, m 求和.将(2.5)式两端同时乘以 {h_{jm}}(t) ,然后也关于 j = 1, 2, \cdots, m 求和.最后再将得到的两个结果式相加,可得 上式中用到结论:对于 u \in H_0^1$$ v \in {H^1}$,可得$((u \cdot \nabla)v, v) = 0$.

$\begin{eqnarray} &&\mathop {\sup }\limits_{0 \le t \le T} ||{u_m}||_{{L^2}}^2 + \mathop {\sup }\limits_{0 \le t \le T} ||{b_m}||_{{L^2}}^2 + 2\int_0^T {||\nabla u||_2^2} {\rm d}t + 2\int_0^T {||\nabla b||_2^2} {\rm d}t + 2\int_0^T {||u||_{{L^{\beta + 1}}}^{\beta + 1}} {\rm d}t\\ & \le& ||{u_0}||_{{L^2}}^2 + ||{b_0}||_{{L^2}}^2. \end{eqnarray}$

$\begin{eqnarray} \mathop {\sup }\limits_{\tau \in R} ||\widehat {{{\tilde b}_m} \cdot \nabla {{\tilde b}_m}}|{|_{{L^2}}} \le C, \end{eqnarray}$

$\begin{eqnarray} \mathop {\sup }\limits_{\tau \in R} ||\widehat {{{\tilde b}_m} \cdot \nabla {{\tilde u}_m}}|{|_{{L^2}}} \le C. \end{eqnarray}$

$$$\mathop {\sup }\limits_{\tau \in R} ||\widehat {|{u_m}{|^{\beta - 1}}{u_m}}( \tau )|{|_{{{\beta + 1} \over \beta }}} \le C.$$$

$$$|{u_m}(0)|{|_{{L^2}}} \le C, ||{u_m}(T)|{|_{{L^2}}} \le C, \quad ||{b_m}( 0 )|{|_{{L^2}}} \le C, ||{b_m}( T )|{|_{{L^2}}} \le C.$$$

$\begin{eqnarray} \int_{ - \infty }^\infty {|\tau {|^{2\gamma }}||{{\widehat {\widetilde u}}_m}||_{{L^2}}^2} {\rm d}\tau & \le& C\int_{ - \infty }^\infty {{{1 + |\tau |} \over {1 + |\tau {|^{1 - 2\gamma }}}}} ||{\widehat {\widetilde u}_m}||_{{L^2}}^2{\rm d}\tau\\ &\le& C\int_{ - \infty }^\infty {||{{\widehat {\widetilde u}}_m}||_{{L^2}}^2} {\rm d}\tau + C\int_{ - \infty }^\infty {{{||{{\widehat {\widetilde u}}_m}( \tau )|{|_{{H^1}}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau\\&& + C\int_{ - \infty }^\infty {{{||{{\widehat {\widetilde u}}_m}( \tau )|{|_{\beta + 1}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau , \end{eqnarray}$

$\begin{eqnarray} \int_{ - \infty }^\infty {|\tau {|^{2\gamma }}||{{\widehat {\widetilde b}}_m}||_{{L^2}}^2} {\rm d}\tau &\le& C\int_{ - \infty }^\infty {{{1 + |\tau |} \over {1 + |\tau {|^{1 - 2\gamma }}}}} ||{\widehat {\widetilde b}_m}||_{{L^2}}^2{\rm d}\tau\\ &\le & C\int_{ - \infty }^\infty {||{{\widehat {\widetilde b}}_m}||_{{L^2}}^2} {\rm d}\tau + C\int_{ - \infty }^\infty {{{||{{\widehat {\widetilde b}}_m}( \tau )|{|_{{H^1}}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau . \end{eqnarray}$

$\begin{eqnarray} && \int_{ - \infty }^\infty {{{||{{\widehat {\widetilde u}}_m}( \tau )|{|_{{H^1}}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau \le {\bigg( {\int_{ - \infty }^\infty {{{{\rm d}\tau } \over {{{(1 + |\tau {|^{1 - 2\gamma }})}^2}}}} }\bigg )^{{1 \over 2}}}{\bigg( {\int_0^T {||{u_m}(\tau )||_{{H^1}}^2{\rm d}\tau } }\bigg )^{{1 \over 2}}} \le C, \end{eqnarray}$

$\begin{eqnarray} &&\int_{ - \infty }^\infty {{{||{{\widehat {\widetilde b}}_m}( \tau )|{|_{{H^1}}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau \le {\bigg( {\int_{ - \infty }^\infty {{{{\rm d}\tau } \over {{{(1 + |\tau {|^{1 - 2\gamma }})}^2}}}} } \bigg)^{{1 \over 2}}}{\bigg( {\int_0^T {||{b_m}(\tau )||_{{H^1}}^2{\rm d}\tau } }\bigg )^{{1 \over 2}}} \le C, \end{eqnarray}$

$\begin{eqnarray} \int_{ - \infty }^\infty {{{||{{\widehat {\widetilde u}}_m}( \tau )|{|_{\beta + 1}}} \over {1 + |\tau {|^{1 - 2\gamma }}}}} {\rm d}\tau & \le &{\bigg( {\int_{ - \infty }^\infty {{{{\rm d}\tau } \over {{{(1 + |\tau {|^{1 - 2\gamma }})}^{{{\beta + 1} \over \beta }}}}}} }\bigg )^{{\beta \over {\beta + 1}}}}{\bigg( {\int_{ - \infty }^\infty {||{{\widehat {\widetilde u}}_m}( \tau )||_{\beta + 1}^{\beta + 1}{\rm d}\tau } } \bigg)^{{1 \over {\beta + 1}}}}\\ &\le &C{\bigg( {\int_{ - \infty }^\infty {||{{\tilde u}_m}( \tau )||_{\beta + 1}^{{{\beta + 1} \over \beta }}{\rm d}\tau } }\bigg )^{{\beta \over {\beta + 1}}}}\\ &\le& C{T^{{{\beta - 1} \over {\beta + 1}}}}{\bigg( {\int_0^T {||{u_m}( \tau )||_{\beta + 1}^{\beta + 1}{\rm d}\tau } }\bigg )^{{1 \over {\beta + 1}}}}. \end{eqnarray}$

$\begin{eqnarray} \int_{ - \infty }^\infty {|\tau {|^{2\gamma }}||{{\widehat {\widetilde u}}_m}||_{{L^2}}^2} {\rm d}\tau \le C , \qquad \int_{ - \infty }^\infty {|\tau {|^{2\gamma }}||{{\widehat {\widetilde b}}_m}||_{{L^2}}^2} {\rm d}\tau \le C . \end{eqnarray}$

$$$\begin{array}{l} u \in {L^\infty }( {0, T;{L^2}( {{{\Bbb R} ^3}} )} ) \cap {L^2}( {0, T;H_0^1( {{{\Bbb R} ^3}} )} ) \cap {L^{\beta + 1}}( {0, T;{L^{\beta + 1}}( {{{\Bbb R} ^3}} )} ), \\ b \in {L^\infty }( {0, T;{L^2}( {{{\Bbb R} ^3}} )} ) \cap {L^2}( {0, T;H_0^1( {{{\Bbb R} ^3}} )} ). \end{array}$$$

将方程组(1.1)中第一个方程乘以${u_t}$,第二个方程乘以${b_t}$,然后分别在${{\Bbb R} ^3}$关于$x$积分,再相加可得

$\begin{eqnarray} & &||{u_t}||_{{L^2}}^2 + ||{b_t}||_{{L^2}}^2 + {1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla u||_{{L^2}}^2 + {1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla b||_{{L^2}}^2 + {1 \over {\beta + 1}}{{\rm d}\over {{\rm d}t}}||u||_{\beta + 1}^{\beta + 1}\\ & = & \int_{{{\Bbb R} ^3}} {b \cdot \nabla b{u_t}} {\rm d}x - \int_{{{\Bbb R} ^3}} {u \cdot \nabla u{u_t}} {\rm d}x - \int_{{{\Bbb R} ^3}} {u \cdot \nabla b{b_t}} {\rm d}x + \int_{{{\Bbb R} ^3}} {b \cdot \nabla u{b_t}} {\rm d}x\\ &\le & \int_{{{\Bbb R} ^3}} {|b \cdot \nabla b||{u_t}|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla u||{u_t}|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla b||{b_t}} |{\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla u||{b_t}} |{\rm d}x . \end{eqnarray}$

$\begin{eqnarray} &&\int_{{{\Bbb R} ^3}} {|b \cdot \nabla b||{u_t}|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla u||{u_t}|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla u||{b_t}} |{\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla b||{b_t}} |{\rm d}x\\ &\le &||u \cdot \nabla u|{|_{{L^2}}}||{u_t}|{|_{{L^2}}} + ||b \cdot \nabla b|{|_{{L^2}}}||{u_t}|{|_{{L^2}}} + ||b \cdot \nabla u|{|_{{L^2}}}||{b_t}|| + ||u \cdot \nabla b|{|_{{L^2}}}||{b_t}|{|_{{L^2}}}\\ &\le& {1 \over 4}||{u_t}||_{{L^2}}^2 + C||u \cdot \nabla u||_{{L^2}}^2 + {1 \over 4}||{u_t}||_{{L^2}}^2 + C||b \cdot \nabla b||_{{L^2}}^2 + {1 \over 4}||{b_t}||_{{L^2}}^2 + C||b \cdot \nabla u||_{{L^2}}^2 \\ &&+ {1 \over 4}||{b_t}||_{{L^2}}^2 + C||u \cdot \nabla b||_{{L^2}}^2\\ &\le & {1 \over 2}||{u_t}||_{{L^2}}^2 + {1 \over 2}||{b_t}||_{{L^2}}^2 + C( {||u \cdot \nabla u||_{{L^2}}^2 + ||b \cdot \nabla b||_{{L^2}}^2 + ||u \cdot \nabla b||_{{L^2}}^2 + ||b \cdot \nabla u||_{{L^2}}^2} )\\ &\le& {1 \over 2}||{u_t}||_{{L^2}}^2 + {1 \over 2}||{b_t}||_{{L^2}}^2 + C( {{I_1} + {I_2} + {I_3} + {I_4}} ). \end{eqnarray}$

$\begin{eqnarray} {I_1}& \le& {( {||u|{|_{{L^4}}}||\nabla u|{|_{{L^4}}}} )^2}\\ &\le& C{( {||u||_{{L^2}}^{{1 \over 4}}||\nabla u||_{{L^2}}^{{3 \over 4}}||\nabla u||_{{L^2}}^{{1 \over 4}}||\Delta u||_{{L^2}}^{{3 \over 4}}} )^2}\\ &\le& C||u||_{{L^2}}^{{1 \over 2}}||\nabla u||_{{L^2}}^2||\Delta u||_{{L^2}}^{{3 \over 2}}\\ &\le & C||u||_{{L^2}}^2||\nabla u||_{{L^2}}^8 + {1 \over 8}||\Delta u||_{{L^2}}^2\\ &\le& C||\nabla u||_{{L^2}}^8 + {1 \over 8}||\Delta u||_{{L^2}}^2. \end{eqnarray}$

$\begin{eqnarray} {I_3} &\le& {( {||u|{|_{{L^4}}}||\nabla b|{|_{{L^4}}}} )^2}\\ &\le& C{( {||u||_{{L^2}}^{{1 \over 4}}||\nabla u||_{{L^2}}^{{3 \over 4}}||\nabla b||_{{L^2}}^{{1 \over 4}}||\Delta b||_{{L^2}}^{{3 \over 4}}} )^2}\\ &\le & C||u||_{{L^2}}^{{1 \over 2}}||\nabla u||_{{L^2}}^{{3 \over 2}}||\nabla b||_{{L^2}}^{{1 \over 2}}||\Delta b||_{{L^2}}^{{3 \over 2}}\\ &\le& C||u||_{{L^2}}^2||\nabla u||_{{L^2}}^6||\nabla b||_{{L^2}}^2 + {1 \over 8}||\Delta b||_{{L^2}}^2\\ &\le& C||\nabla u||_{{L^2}}^8 + C||\nabla b||_{{L^2}}^8 + {1 \over 8}||\Delta b||_{{L^2}}^2. \end{eqnarray}$

$$${I_2} \le C||\nabla b||_{{L^2}}^8 + {1 \over 8}||\Delta b||_{{L^2}}^2,$$$

$$${I_4} \le C||\nabla b||_{{L^2}}^8 + C||\nabla u||_{{L^2}}^8 + {1 \over 8}||\Delta u||_{{L^2}}^2.$$$

$\begin{eqnarray} & &{1 \over 2} ||{u_t}||_{{L^2}}^2 + {1 \over 2} ||{b_t}||_{{L^2}}^2 + {1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla u||_{{L^2}}^2 + {1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla b||_{{L^2}}^2 + {1 \over {\beta + 1}}{{\rm d}\over {{\rm d}t}}||u||_{\beta + 1}^{\beta + 1}\\ &\le& C||\nabla u||_{{L^2}}^8 + C||\nabla b||_{{L^2}}^8 + {1 \over 4}||\Delta u||_{{L^2}}^2 + {1 \over 4}||\Delta b||_{{L^2}}^2. \end{eqnarray}$

$\begin{eqnarray} & &{1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla u||_{{L^2}}^2 + {1 \over 2}{{\rm d}\over {{\rm d}t}}||\nabla b||_{{L^2}}^2 + ||\Delta u||_{{L^2}}^2 + ||\Delta b||_{{L^2}}^2 + \int_{{{\Bbb R} ^3}} {|u{|^{\beta - 1}}|\nabla u{|^2}} {\rm d}x \\ &&+ {{\beta - 1} \over 4}\int_{{{\Bbb R} ^3}} {|u{|^{\beta - 3}}|\nabla |u{|^2}} {|^2}{\rm d}x\\ & = & \int_{{{\Bbb R} ^3}} {u \cdot \nabla u\Delta u} {\rm d}x - \int_{{{\Bbb R} ^3}} {b \cdot \nabla b\Delta u} {\rm d}x + \int_{{{\Bbb R} ^3}} {u \cdot \nabla b\Delta b} {\rm d}x - \int_{{{\Bbb R} ^3}} {b \cdot \nabla b\Delta b{\rm d}x}\\ &\le& \int_{{{\Bbb R} ^3}} {|u \cdot } \nabla u||\Delta u|{\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla b||\Delta u|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla b||\Delta b|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla u||\Delta b|} {\rm d}x. \end{eqnarray}$

$\begin{eqnarray} &&\int_{{{\Bbb R} ^3}} {|u \cdot } \nabla u||\Delta u|{\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla b||\Delta u|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla b||\Delta b|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla u||\Delta b|} {\rm d}x\\ &\le& {2 \over 8}||\Delta u||_{{L^2}}^2 + {2 \over 8}||\Delta b||_{{L^2}}^2 + C( {||u \cdot \nabla u||_{{L^2}}^2 + ||u \cdot \nabla b||_{{L^2}}^2 + ||b \cdot \nabla b||_{{L^2}}^2 + ||b \cdot \nabla u||_{{L^2}}^2} ). \qquad \end{eqnarray}$

$\begin{eqnarray} &&\int_{{{\Bbb R} ^3}} {|u \cdot } \nabla u||\Delta u|{\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla b||\Delta u|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|u \cdot \nabla b||\Delta b|} {\rm d}x + \int_{{{\Bbb R} ^3}} {|b \cdot \nabla u||\Delta b|} {\rm d}x\\ &\le& C( {||\nabla u||_{{L^2}}^8 + ||\nabla b||_{{L^2}}^8} ) + {3 \over 8}||\Delta u||_{{L^2}}^2 + {3 \over 8}||\Delta b||_{{L^2}}^2. \end{eqnarray}$

$\begin{eqnarray} &&{1 \over 2} ||{u_t}||_{{L^2}}^2 + {1 \over 2} ||{b_t}||_{{L^2}}^2 + {{\rm d}\over {{\rm d}t}}||\nabla u||_{{L^2}}^2 + {{\rm d}\over {{\rm d}t}}||\nabla b||_{{L^2}}^2 + {1 \over {\beta + 1}}{{\rm d}\over {{\rm d}t}}||u||_{\beta + 1}^{\beta + 1}\\ & &{\rm{ + }}{3 \over 8}||\Delta u||_{{L^2}}^2{\rm{ + }}{3 \over 8}||\Delta {\rm{b}}||_{{L^2}}^2+ \int_{{{\Bbb R} ^3}} {|u{|^{\beta - 1}}|\nabla u{|^2}} {\rm d}x + {{\beta - 1} \over 4}\int_{{{\Bbb R} ^3}} {|u{|^{\beta - 3}}|\nabla |u{|^2}} {|^2}{\rm d}x \\ &\le& C( {||\nabla u||_{{L^2}}^8 + ||\nabla b||_{{L^2}}^8} )\le C{( {||\nabla u||_{{L^2}}^2 + ||\nabla b||_{{L^2}}^2 + ||u||_{\beta + 1}^{\beta + 1}} )^4}. \end{eqnarray}$

$\begin{eqnarray} {{\rm d}\over {{\rm d}t}}I( t ) \le C{( {I( t )} )^4}, \qquad I( t ) \le {1 \over {{{( {I{{( 0 )}^{ - 3}} - Ct} )}^{{1 \over 3}}}}}, \end{eqnarray}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Li D Q , Qin T H . Physics and Partial Differential Equations. Beijing: Higher Education Press, 1997

Cabannes H , Sfeir A A . Theoretical Magnetofluiddynamics. New York: Academic Press, 1970

Lifshitz E M , Pitaevskii L P . Electrodynamics of Continuous Media. New York: Pergamon Press, 1984

Duvaut G , Lions J L .

Inéquations en thermoélasticité et magnétohydrodynamique

Arch Ration Mech Anal, 1972, 46: 241- 279

Sermange M , Temam R .

Some mathematical questions related to the MHD equations

Comm Pure Appl Math, 1983, 36 (5): 635- 664

Wu J H. The 2D magnetohydrodynamic equations with partial or fractional dissipation//Yang L, Yau S-T. Morningside Lectures in Mathematics. Somerville, MA: International Press, 2017

Hsiao L . Quasilinear Hyperbolic Systems and Dissipative Mechanisms. Singapore: World Scientific, 1997

Cai X , Jiu Q .

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

Journal of Mathematical Analysis and Applications, 2008, 343 (2): 799- 809

Zhang Z , Wu X , Lu M .

On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping

Journal of Mathematical Analysis and Applications, 2011, 377 (1): 414- 419

Zhou Y .

Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping

Applied Mathematics Letters, 2012, 25 (11): 1822- 1825

Temam R . Navier-Stokes Equations, Theory and Numerical Analysis. Amsterdam: North-Holland, 1977

/

 〈 〉