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

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 $(\Vert {\rho }_0{\Vert }_{L^{\infty }}+1)[\Vert {\rho }_0{\Vert }_{L^3}+\Vert {\rho }_0{\Vert }_{L^{\infty }}+1{)}^2(\Vert \sqrt{{\rho }_0}{u}_0{\Vert }_{L^2}^2+\Vert b_0{\Vert }_{L^2}^2\left)\right][\Vert \nabla u_0{\Vert }_{L^2}^2+(\Vert {\rho }_0{\Vert }_{L^{\infty }}+1)(\Vert \sqrt{{\rho }_0}{E}_0{\Vert }_{L^2}^2+\Vert \nabla b_0{\Vert }_{L^2}^2\left)\right]$ 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].

arXiv commentThis is a final version accepted by Journal of Differential Equations