On Classical Solutions of the Compressible Magnetohydrodynamic Equations with Vacuum

Abstract

In this paper, we consider the three-dimensional compressible isentropic magnetohydrodynamic equations with infinite electric conductivity. The existence of unique local classical solutions is established when the initial data are arbitrarily large, contain vacuum, and satisfy some initial layer compatibility condition. The initial mass density need not be bounded away from zero and may vanish in some open set or decay at infinity. Moreover, we prove that the $L^\infty$ norm of the deformation tensor of velocity gradients controls the possible blow-up (see [O. Rozanova, “Blow Up of Smooth Solutions to the Barotropic Compressible Magnetohydrodynamic Equations with Finite Mass and Energy,” in Hyperbolic Problems: Theory, Numerics and Applications, AMS, Providence, RI, 2009, pp. 911--917], [Z. Xin, Comm. Pure Appl. Math., 51 (1998), pp. 229--240]) for classical (or strong) solutions, which means that if a solution of the compressible magnetohydrodynamic equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the deformation tensor as the critical time approaches. Our criterion (see (1.12)) is the same as Ponce's criterion for three-dimensional incompressible Euler equations [G. Ponce, Comm. Math. Phys., 98 (1985), pp. 349--353] and Huang, Li, and Xin's criterion for the three-dimensional compressible Navier--Stokes equations [X. Huang, J. Li, and Z. Xin, Comm. Math. Phys., 301 (2011), pp. 23--35].