On the Global Existence of Classical Solutions for Compressible Magnetohydrodynamic Equations

This paper deals with the Cauchy problem of compressible magnetohydrodynamic equations in the whole space ℝ 3 . We show that if, in addition, the conservation law of the total mass is satisfied (i.e., ρ 0 ∈ L 1 ), then the global existence theorem with small density and L 3 -norm of H 0 holds for any γ > 1. It is worth mentioning that the initial velocity can be arbitrarily large and the initial vacuum is allowed. Thus, the result obtained particularly extends the one due to Li et al. (SIAM J. Math. Anal., 45:1356–1387, 2013 ), where the global well-posedness of classical solutions with small energy was proved.