The lifespan of strong solutions to the compressible MHD equations with entropy transport in the presence of vacuum

TL;DR

This paper studies the finite-time blow-up of strong solutions to the compressible MHD equations with entropy transport, providing conditions for singularity formation and explicit lifespan estimates, extending previous results to free boundary problems.

Contribution

It establishes new blow-up criteria for solutions with vacuum regions and extends lifespan estimates to free boundary scenarios in compressible MHD.

Findings

Finite-time blow-up occurs when initial density vanishes in a vacuum region with a non-trivial magnetic field.

Local well-posedness of strong solutions is proved for bounded domains.

Explicit lifespan bounds are derived based on boundary expansion analysis.

Abstract

In this paper, we investigate the finite time blow-up of strong solutions to the compressible magnetohydrodynamic (MHD) system (without magnetic diffusion) coupled with entropy transport, and derive an upper bound for the lifespan of such solutions. We first establish the local well-posedness of strong solutions for bounded domains and study the mechanism of finite-time singularity formation in the 2D radially symmetric case and 3D cylindrically symmetric case. We prove that if the initial density vanishes in an interior region containing the origin and the magnetic field is non-trivial within this vacuum region, the strong solution must blow up in finite time. These results generalize and improve the previous results of Huang-Xin-Yan [Math. Ann. 392 (2025) 2365-2394] for the compressible isentropic MHD equations. Significantly, we extend this blow-up result to the free boundary problem. Our analysis of the boundary's expansion allows us to explicitly estimate the maximum lifespan of the solution.