Global well-posedness of the three-dimensional non-isentropic compressible magnetohydrodynamic equations under a scaling-invariant smallness condition

Abstract

We consider the Cauchy problem of the non-isentropic compressible magnetohydrodynamic equations in ${\mathbb{R}}^3$ with far-field vacuum. By deriving delicate energy estimates and exploiting the intrinsic structure of the system, we establish the global existence and uniqueness of strong solutions provided that the scaling-invariant quantity \begin{align*} (1+\barρ+\tfrac{1}{\barρ}) [\|ρ_{0}\|_{L^{3}}+ ( \barρ^{2}+\barρ)( \| \sqrt{ρ_{0}}u_{0}\|_{L^{2}}^{2}+\| b_{0}\|_{L^{2}}^{2}) ] [\|\nabla u_{0}\|_{L^{2}}^{2}+(\barρ+1)\|\sqrt{ρ_{0}} θ_{0}\|_{L^{2}}^{2}+\| \nabla b_{0}\|_{L^{2}}^{2}+\| b_{0}\|_{L^{4}}^{4} ] \end{align*} is sufficiently small, where $\bar{\ddot{I}}$ denotes the essential supremum of the initial density. Our result may be regarded as an improved version compared with that of Liu and the second author (J. Differential Equations 336 (2022), pp. 456--478) in the sense that an artificial condition $3\hat{I}¼>\hat{I}»$ on the viscosity coefficients is removed. In particular, we provide a new scaling-invariant quantity regarding the initial data.

19pages