Global well-posedness of the three-dimensional non-isentropic compressible magnetohydrodynamic equations under a scaling-invariant smallness condition
Lin Xu, Xin Zhong
TL;DR
This work addresses the global well-posedness of the 3D non-isentropic compressible MHD equations with vacuum by introducing a scaling-invariant smallness condition on the initial data. The authors develop refined energy estimates and a bootstrap framework to obtain uniform-in-time bounds, notably controlling the density in $L^3$ and the magnetic-field interactions, and they remove the artificial $3\mu>\lambda$ restriction. A Serrin-type blow-up criterion is then used to extend local strong solutions to global ones. The result advances the theory by establishing global strong solutions under a natural, scaling-invariant framework and highlights the pivotal role of the magnetic field in the scaling analysis.
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 $\barρ$ 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 $3μ>λ$ on the viscosity coefficients is removed. In particular, we provide a new scaling-invariant quantity regarding the initial data.