On a degenerate boundary value problem to relativistic magnetohydrodynamics with a general pressure law

TL;DR

This paper establishes the existence and uniqueness of solutions for a degenerate boundary value problem in relativistic magnetohydrodynamics with a general pressure law, addressing challenges posed by degeneracy and magnetic effects.

Contribution

It introduces a novel analysis of a relativistic MHD boundary value problem with a general equation of state and magnetic effects, previously unexplored in this context.

Findings

Proved local existence and uniqueness of classical solutions.

Reduced complex MHD system to a simplified form for analysis.

Applied iteration method to handle degeneracy along the sonic curve.

Abstract

This work is concerned with establishing the existence and uniqueness of the solution to a mixed-type degenerate boundary value problem for a relativistic magnetohydrodynamics system. We first consider a full relativistic magnetohydrodynamics system and reduce it to a simplified form under the assumption that the magnetic field vector is orthogonal to the velocity vector. We consider a boundary value problem for the steady part of this reduced system where the boundary data is prescribed on a sonic boundary and a characteristic curve. Here the main difficulty is the consideration of a relativistic system, with a general equation of state while considering the magnetic field effects as well, which we believe has never been analyzed before in the context of the analytical study of sonic-supersonic flows. Also, the degeneracy of the governing equations along the sonic curve is a crucial challenge. However, we employ the iteration method used in the work of Li and Hu \cite{li2019degenerate} to prove the existence and uniqueness of a local classical supersonic solution in the partial hodograph plane first and finally, we recover a local smooth solution to the boundary value problem in the physical plane by applying an inverse transformation.