MHD equations with regularity in one direction
Summary: We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say, \(\partial_3\mathbf{u}\in L^p(0,T;L^q(\mathbb {R}^3))\), with \(2/p+3/q=\gamma\in [1,3/2)\), \(3/\gamma \leq q\leq 1/(\gamma -1)\), then the solution is in fact smooth. This improves previous results greatly.
scientific article; zbMATH DE number 6420097
scientific article