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

Abstract

We study the initial-boundary value problem for 1D compressible magnetohydrodynamics equations of viscous nonresistive 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 [Nonlinearity 30, 3587–3612 (2017)]. The 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 L2- and H1-norm, respectively, with large initial data.