Global weak solutions and long time behavior for 1D compressible MHD equations without resistivity

TL;DR

This paper establishes the existence, uniqueness, and long-term decay behavior of weak solutions for 1D compressible non-resistive MHD equations, advancing understanding of their stability and dynamics over time.

Contribution

It constructs global weak solutions for 1D compressible MHD without resistivity, proves their uniqueness, and analyzes their exponential decay over time.

Findings

Existence of global weak solutions was established.

Uniqueness of solutions was proved.

Solutions decay exponentially in L^2 and H^1 norms.

Abstract

We study the initial-boundary value problem for 1D compressible MHD equations of viscous non-resistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are constructed by approximation of global regular solutions, the existence of which has recently been obtained by Jiang and Zhang in [17]. Uniqueness of weak solutions is also proved as a consequence of Lipschitz continuous dependence on the initial data. Furthermore, long time behavior for global solutions is investigated. Specifically, based on the uniform-in-time bounds of the density from above and below away from zero, together with the structure of the equations, we show the exponential decay rate in L^2- and H^1-norm respectively, with initial data of arbitrarily large.