Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity

TL;DR

This paper proves that under certain conditions, the entropy in 3D compressible heat-conducting magnetohydrodynamic flows with vacuum at infinity remains bounded, using advanced energy estimates and iteration techniques.

Contribution

It establishes the boundedness of entropy and regularity of velocity and temperature for the first time in this setting, extending previous methods to MHD flows.

Findings

Entropy remains bounded under suitable initial decay conditions.

Velocity and temperature regularities are propagated over time.

New mathematical techniques for entropy bounds are developed.

Abstract

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In particular, it is unknown whether the entropy remains its boundedness. In the present paper, we investigate the Cauchy problem to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity only. We show that the uniform boundedness of the entropy and the $L^2$ regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. The main tools are based on singularly weighted energy estimates and De Giorgi type iteration techniques developed by Li and Xin (arXiv:2111.14057) for the 3D full compressible Navier-Stokes system. Some new mathematical techniques and useful estimates are developed to deduce the lower and upper bounds on the entropy.