Finite time blowup of strong solutions to the two dimensional MHD equations

Abstract

Whether the smooth solution of the multi-dimensional viscous compressible fluids will blow-up in finite time has always been a chanllenging problem. In the recent work\cite{FM}, Merle et al. proved that there are smooth solutions to the 2D radially symmetric compressible Navier-Stokes equations which will inevitably form shell singularities in finite time.\\ \indent In this article, we first prove the existence of local strong solutions that allow vacuum for the two-dimensional viscous compressible MHD equations on bounded domains without magnetic diffusion. Furthermore, it is shown that if the initial data are radial symmetric and its vacuum set contains a ball centered at the origin where the total magnetic field is non-trivial, then the radial symmetric strong solution to the initial boundary value problem will definitely blow up in finite time. This is the first example for the formation of finite time singularity of strong solutions that allows interior vacuum of a viscous compressible fluid.