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

TL;DR

This paper proves the global existence of classical solutions to 3D compressible magnetohydrodynamic equations with large initial data and vacuum, under certain smallness conditions on specific energy and magnetic field norms.

Contribution

It extends previous results by allowing large initial energy and no restrictions on initial density and velocity, improving the understanding of MHD solutions with vacuum.

Findings

Global classical solutions exist under small energy and magnetic field norms.

Initial energy can be large as b7 b7; the result applies even with vacuum.

No restrictions on initial density and velocity.

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.