On Echoes in Magnetohydrodynamics with Magnetic Dissipation

TL;DR

This paper investigates the long-term behavior of magnetohydrodynamic equations with magnetic dissipation, revealing the existence of special solutions and resonances that lead to norm inflation and blow-up phenomena.

Contribution

It introduces explicit global low-frequency solutions and analyzes resonance-induced blow-up in the linearized MHD equations with magnetic dissipation.

Findings

Existence of explicit global low-frequency solutions.

Resonances called echoes cause norm inflation.

Resonances lead to infinite time blow-up in Sobolev regularity.

Abstract

We study the long time asymptotic behavior of the inviscid magnetohydrodynamic equations with magnetic dissipation near a combination of Couette flow and a constant magnetic field. Here we show that there exist nearby explicit global in time low frequency solutions, which we call waves. Moreover, the linearized problem around these waves exhibits resonances under high frequency perturbations, called echoes, which result in norm inflation Gevrey regularity and infinite time blow-up in Sobolev regularity.