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

TL;DR

This paper proves the global existence and uniqueness of strong solutions for 3D non-isentropic compressible MHD equations under a new smallness condition that is scaling-invariant, removing previous restrictions on viscosity coefficients.

Contribution

It introduces a new scaling-invariant smallness condition for initial data ensuring global well-posedness, improving upon prior results by removing viscosity coefficient restrictions.

Findings

Established global existence and uniqueness of strong solutions.

Derived a new scaling-invariant initial data condition.

Removed the artificial viscosity coefficient restriction.

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+\bar{\rho}+\tfrac{1}{\bar{\rho}}) [\|\rho_{0}\|_{L^{3}}+ ( \bar{\rho}^{2}+\bar{\rho})( \| \sqrt{\rho_{0}}u_{0}\|_{L^{2}}^{2}+\| b_{0}\|_{L^{2}}^{2}) ] [\|\nabla u_{0}\|_{L^{2}}^{2}+(\bar{\rho}+1)\|\sqrt{\rho_{0}} \theta_{0}\|_{L^{2}}^{2}+\| \nabla b_{0}\|_{L^{2}}^{2}+\| b_{0}\|_{L^{4}}^{4} ] \end{align*} is sufficiently small, where $\bar{\rho}$ 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 $3\mu>\lambda$ on the viscosity coefficients is removed. In particular, we provide a new scaling-invariant quantity regarding the initial data.