Weak solutions to a compressible viscous non-resistive MHD equations with general boundary data

TL;DR

This paper proves the existence of global weak solutions for a compressible viscous non-resistive MHD system with general boundary conditions, extending previous results and establishing a weak-strong uniqueness principle.

Contribution

It extends prior work by handling general boundary data and proves a weak-strong uniqueness principle for the system.

Findings

Existence of global-in-time weak solutions under general boundary conditions.

Extension of previous results to non-homogeneous boundary data.

Establishment of weak-strong uniqueness principle.

Abstract

This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of global-in-time weak solutions with finite energy initial data. The present result extends considerably the previous work by Li and Sun [\emph{J. Differential Equations.}, 267 (2019), pp. 3827-3851], where the homogeneous Dirichlet boundary condition for velocity field is treated. The proof leans on the specific mathematical structure of equations and the recently developed theory of open fluid systems. Furthermore, we establish the weak-strong uniqueness principle, namely a weak solution coincides with the strong solution on the lifespan of the latter provided they emanate from the same initial and boundary data. This basic property is expected to be useful in the study of convergence of numerical solutions.