Global well-posedness of planar MHD system without heat conductivity

TL;DR

This paper proves the global existence and uniqueness of strong solutions for the planar MHD system without heat conduction, accommodating vacuum states and density discontinuities, using novel energy estimates and coupling techniques.

Contribution

It introduces a new coupling structure between velocity and magnetic field to establish well-posedness without relying on entropy-type inequalities.

Findings

Global well-posedness for large initial data with vacuum

Allows density discontinuities and interior vacuum

Develops new coupling method for dissipative estimates

Abstract

In this paper, we consider the Cauchy problem to the planar magnetohydrodynamics (MHD) system with both constant viscosity and constant resistivity but without heat conductivity. Global well-posedness of strong solutions in the presence of natural far field vacuum, due to the finiteness of the mass, is established for any large initial data of suitable smoothness. Density discontinuity and interior vacuum of either point-like or piecewise-like are also allowed. Technically, the entropy-type energy inequality, which although is commonly used as a basic tool in existing literature on the planar MHD system, is not workable in the current paper, as it is not consistent with the far field vacuum. Instead, besides making full use of advantages of the effective viscous flux as in \cite{LJK1DNONHEAT,LIJLIM2022,LIXINADV}, a new coupling structure, between the longitudinal velocity $u$ and the transversal magnetic field $\bm h$, is exploited to recover the dissipative estimate on $\bm h$.