Remarks on the regularity criteria for generalized MHD equations
The Cauchy problem to the 3D generalized MHD equations is studied in \(\{x\in{\mathbb R}^3,t>0\}\). \[ \begin{aligned} &\frac{\partial u}{\partial t}+(-\Delta)^\alpha u+u\cdot\nabla u -b\cdot\nabla b+\nabla p=0,\quad \text{div}\,u=0,\\ & \frac{\partial b}{\partial t}+(-\Delta)^\beta b +u\cdot\nabla b -b\cdot\nabla u=0,\quad \text{div}\,b=0,\\ &u(x,0)=u_0(x),\quad b(x,0)=b_0(x). \end{aligned} \] Here \(u\) is the velocity field, \(b\) is the magnetic field, \(p\) is the pressure, \(\alpha,\beta>0\), \((-\Delta)^\alpha\) is the fractional power of Laplacian. It is proved that if \(u_0,b_0\in H^1({\mathbb R}^3)\) and \((u,b)\) is a weak solution with \(\nabla u\) integrable in a certain Morrey space then \((u,b)\) is smooth on \((0,T)\).
generalized magneto-hydrodynamics equations
scientific article; zbMATH DE number 5833515