Entropy-bounded solutions to the Cauchy problem of compressible planar non-resistive magnetohydrodynamics equations with far field vacuum

TL;DR

This paper studies the behavior of entropy in solutions to the compressible magnetohydrodynamics equations with vacuum at infinity, establishing conditions under which entropy remains bounded over time.

Contribution

It provides the first analysis of entropy behavior near vacuum regions for this model, extending previous existence results with new a priori estimates.

Findings

Entropy remains bounded over finite time under certain initial conditions.

Boundedness of entropy propagates if initial vacuum decays sufficiently slowly.

The results apply to solutions with vacuum at far fields, enhancing understanding of the model's physical behavior.

Abstract

We investigate the Cauchy problem to the compressible planar non-resistive magnetohydrodynamic equations with zero heat conduction. The global existence of strong solutions to such a model has been established by Li and Li (J. Differential Equations 316: 136--157, 2022). However, to our best knowledge, so far there is no result on the behavior of the entropy near the vacuum region for this model. The main novelty of this paper is to give a positive response to this problem. More precisely, by a series of a priori estimates, especially the singular type estimates, we show that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density.