Global Solutions of Three-dimensional Inviscid MHD Fluids with Velocity Damping in Horizontally Periodic Domains

TL;DR

This paper proves the existence of unique global solutions with exponential decay for three-dimensional inviscid MHD fluids with velocity damping in horizontally periodic domains, extending previous 2D results.

Contribution

It introduces a novel two-layer energy method to handle nonlinear terms in 3D MHD equations, establishing global existence and decay results.

Findings

Global solutions exist and are unique.

Solutions decay exponentially over time.

New energy method overcomes 3D nonlinear challenges.

Abstract

The \emph{two-dimensional} (2D) existence result of global(-in-time) solutions for the motion equations of incompressible, inviscid, non-resistive magnetohydrodynamic (MHD) fluids with velocity damping had been established in [Wu--Wu--Xu, SIAM J. Math. Anal. 47 (2013), 2630--2656]. This paper further studies the existence of global solutions for the \emph{three-dimensional} (a dimension of real world) initial-boundary value problem in a horizontally periodic domain with finite height. Motivated by the multi-layers energy method introduced in [Guo--Tice, Arch. Ration. Mech. Anal. 207 (2013), 459--531], we develop a new type of two-layer energy structure to overcome the difficulty arising from three-dimensional nonlinear terms in the MHD equations, and thus prove the initial-boundary value problem admits a unique global solution. Moreover the solution has the exponential decay-in-time around some rest state. Our two-layer energy structure enjoys two features: (1) the lower-order energy (functional) can not be controlled by the higher-order energy. (2) under the \emph{a priori} smallness assumption of lower-order energy, we first close the higher-order energy estimates, and then further close the lower-energy estimates in turn.