Global strong solution for 3D compressible heat-conducting magnetohydrodynamic equations revisited

TL;DR

This paper establishes the global existence and uniqueness of strong solutions for 3D compressible heat-conducting magnetohydrodynamic equations with vacuum, under a novel smallness condition independent of initial data norms, improving previous results.

Contribution

It introduces a new smallness criterion ensuring global solutions, advancing the understanding of MHD equations with vacuum in three dimensions.

Findings

Proves global strong solutions exist under specific smallness conditions.

Smallness condition is independent of initial data norms.

Improves upon previous results in the field.

Abstract

We revisit the 3D Cauchy problem of compressible heat-conducting magnetohydrodynamic equations with vacuum as far field density. By delicate energy method, we derive global existence and uniqueness of strong solutions provided that $(\|\rho_0\|_{L^\infty}+1)\big[\|\rho_0\|_{L^3}+ \|\rho_0\|_{L^\infty}+1)^2\big(\|\sqrt{\rho_0}u_0\|_{L^2}^2 +\|b_0\|_{L^2}^2\big)\big]\big[\|\nabla u_0\|_{L^2}^2+(\|\rho_0\|_{L^\infty}+1)\big(\|\sqrt{\rho_0}E_0\|_{L^2}^2+\|\nabla b_0\|_{L^2}^2\big)\big]$ is properly small. In particular, the smallness condition is independent of any norms of the initial data. This work improves our previous results [18, 19].