Global Existence for a Class of Large Solutions to Three-Dimensional Compressible Magnetohydrodynamic Equations with Vacuum

Abstract

In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $((\gamma-1)^{\frac{1}{9}}+\nu^{-\frac{1}{4}})E_0$ is suitably small, a result on the existence of the global classical solution is established, where $\gamma$, $\nu$, and $E_0$ represent the adiabatic exponent, resistivity coefficient, and initial energy, respectively. The solution in this paper is exactly a Nishida--Smoller type solution due to the fact that initial energy $E_{0}$ can be large as long as $\gamma$ is close to 1 and $\nu$ is suitably large. Our result also improves the one established by [H. L. Li, X. Y. Xu, and J. W. Zhang, SIAM J. Math. Anal., 45 (2013), pp. 1356--1387], where with small initial energy but possibly large oscillations, the existence of the classical solution was proved.