带阻尼的磁流体方程组的整体适定性

在 [Zujin Zhang, Chupeng Wu, Zheng-an Yao, Remarks on global regularity for the 3D MHD system with damping, Applied Mathematics and Computation, 333 (2018), 1—7] 中, 我们考虑带阻尼的磁流体方程组 $$\bee\label{MHD_damping} \sedd{\ba{ll} \p_t\bbu+(\bbu\cdot\n)\bbu -(\bbb\cdot\n)\bbb -\lap\bbu +|\bbu|^{\al-1}\bbu +\n\pi=\bf{0},\\ \p_t\bbb+(\bbu\cdot\n)\bbb -(\bbb\cdot\n)\bbu -\lap\bbb +|\bbb|^{\beta-1}\bbb =\bf{0},\\ \n\cdot\bbu=\n\cdot\bbb=0,\\ \bbu|_{t=0}=\bbu_0,\quad \bbb|_{t=0}=\bbb_0, \ea} \eee$$ 并证明了如果 $$\bee\label{thm:1} 3\leq \al\leq \f{27}{8},\quad \be\geq 4; \eee$$ $$\bee\label{thm:2} \f{27}{8}<\al\leq\f{7}{2},\quad \be\geq \f{7}{2\al-5}; \eee$$ $$\bee\label{thm:3} \f{7}{2}<\al<4,\quad \be\geq \f{5\al+7}{2\al}; \eee$$ $$\bee\label{thm:4} \al\geq 4,\quad \be\geq 1. \eee$$ 那么 \eqref{MHD_damping} 有一个唯一的整体强解. 主要想法有两个: 一是阻尼越强, 整体适定性应该更好做; 二是速度场如果足够好, 那么磁场可不要阻尼.