Weak Solutions to a Compressible Viscous Non-resistive MHD Equations with General Boundary Data

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 (J Differ Equ 267:3827–3851, 2019), 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.