We assume that Ω is either the whole space ${\mathbb{R}}^3$ or a half-space or a smooth bounded or exterior domain in ${\mathbb{R}}^3$, $T>0$ and $(\mathbf{u},\mathbf{b},p)$ is a suitable weak solution of the MHD equations in $\mathrm{\Omega }\times (0,T)$. We show that $({\mathbf{x}}_0,t_0)\in \mathrm{\Omega }\times (0,T)$ is a regular point of the solution $(\mathbf{u},\mathbf{b},p)$ if the limit inferior (for $t\to t_0-$) of the sum of the $L^3$–norms of u and b over an arbitrarily small ball $B_{\rho }({\mathbf{x}}_0)$ is less than infinity.

中文翻译：

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MHD方程合适弱解的局部正则性的新充分条件

我们假设Ω是整个空间 ${\mathbb{[R}}^3$ 或位于 ${\mathbb{[R}}^3$， $Ť>0$ 和 $（\mathbf{ü}，\mathbf{b}，p）$ 是MHD方程的适当弱解 $\mathrm{\Omega }\times （0，Ť）$。我们证明$（{\mathbf{X}}_0，{Ť}_0）\in \mathrm{\Omega }\times （0，Ť）$ 是解决方案的常规点 $（\mathbf{ü}，\mathbf{b}，p）$ 如果极限低于 $Ť\to {Ť}_0-$）的总和 ${\mathrm{大号}}^3$–在任意小的球上的u和b的范数${乙}_{\rho }（{\mathbf{X}}_0）$ 小于无穷大。