Global classical solution to 3D compressible magnetohydrodynamic equations with large initial data and vacuum

Abstract

In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $(\gamma-1)^{\frac{1}{6}}E_{0}^{\frac{1}{2}}$, together with the $\|H_{0}\|_{L^{2}}$, is suitably small, a result on the existence of global classical solutions is obtained. It should be pointed out that the initial energy $E_{0}$ except the $L^{2}$- norm of $H_{0}$ can be large as $\gamma$ goes to 1, and that throughout the proof of the theorem in the present paper, we make no restriction upon the initial data $(\rho_{0},u_{0})$. Our result improves the one established by Li-Xu-Zhang in \cite{H.L. L}, where, with small initial engergy, the existence of classical solution was proved.