Singularity formation to the Cauchy problem of the two-dimensional non-baratropic magnetohydrodynamic equations without heat conductivity

We study the breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists globally if the gradient of the velocity and the pressure satisfy $\|\nabla\mathbf{u}\|_{L^{1}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In particular, the criterion is independent of the magnetic field and the vacuum in the solutions is allowed.