On the ideal compressible magnetohydrodynamic equations

In this paper, we consider the short time strong solutions to the ideal compressible magnetohydrodynamic equations with initial vacuum, where the velocity field satisfies the Navier-slip condition. Inspired by Kato and Lax's idea, we use the contraction mapping argument to prove the local existence. Moreover, under the Navier-slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L ∞ norm of the deformation tensor D ( u ) .