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

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].