Blow-up Criterion For Two-dimensional Magnetohydrodynamic Equations With Vacuum

This paper studies blow-up criterion for strong solutions to two-dimensional(2D)magnetohydrodynamic(MHD)equations for the initial density contains vacuum.On the one hand,we establish a blow-up criterion which depends only on the space weighted density for the strong solutions to the Cauchy problem of 2D incompressible MHD equations.On the other hand,we prove a blow-up criterion which depends only on the velocity for the strong solutions to the initial boundary value problem of 2D full compressible MHD equations.First,we consider the 2D nonhomogeneous incompressible MHD equations where t is time,? = ?(x,t),u=(u1,u2)(x,t),H=(H1,H2)(x,t),p=p(x,t),denote the density,velocity,magnetic and pressure of the fluid,respectively.We consider the Cauchy problem with(?,u,H)vanishes at infinity and the initial conditions:for given initial data P0,00 and H0.If the initial density contains vacuum(may have compact support),let T*<? be the maximal time of existence,we prove that where ?>0 is a constant and x is the space weighted function.Our result shows that the blow-up criterion depends only on the space weighted density,and is independent of the magnetic and velocity.In particular,this criterion is the same as those of the Navier-Stokes equations.Next,we consider the initial boundary value problem of the full compressible MHD equations on smooth bounded domain ?(?)R2,with the initial conditions(?,u,?,H)(x,t = 0)=(?0,u0,?0,H0),and the boundary conditions u = 0,??/?n = 0,H = 0,on??.Here,?,u,?,H,P represent,respectively,the fluid density,velocity,absolute temperature and magnetic field and pressure.If the initial density contains vacuum,let T*<? be the maximal time of existence,we prove that(?)This implies that the criterion is independent of the magnetic and temperature,and depends only on the divergence of velocity.Moreover,our result is also the same as those of the Navier-Stokes equations.This paper obtains the blow-up criterion for strong solutions to 2D MHD equations with vacuum,which refines some known results.Furthermore,we show that the magnetic field does not play the key role in the study of the mechanism of blow-up.