The equations of extended magnetohydrodynamics

TL;DR

This paper investigates the Cauchy problem for extended magnetohydrodynamics (XMHD), demonstrating well-posedness of the initial value problem for both incompressible and compressible cases with viscosity, and analyzing inertial wave propagation.

Contribution

It recasts XMHD as a symmetric hyperbolic-parabolic system, proving local well-posedness and minimal regularity requirements for the initial value problem.

Findings

XMHD can be formulated as a well-posed symmetric hyperbolic-parabolic system.

Local in time solutions exist for both incompressible and compressible XMHD.

Inertial waves emerge and propagate within the XMHD framework.

Abstract

Extended magnetohydrodynamics (XMHD) is a fluid plasma model generalizing ideal MHD by taking into account the impact of Hall drift effects and the influence of electron inertial effects. XMHD has a Hamiltonian structure which has received over the past ten years a great deal of attention among physicists, and which is embodied by a non canonical Poisson algebra on an infinite-dimensional phase space. XMHD can alternatively be formulated as a nonlinear evolution equation. Our aim here is to investigate the corresponding Cauchy problem. We consider both incompressible and compressible versions of XMHD with, in the latter case, some additional bulk (fluid) viscosity. In this context, we show that XMHD can be recast as a well-posed symmetric hyperbolic-parabolic system implying pseudo-differential operators of order zero acting as coefficients and source terms. Along these lines, we can solve locally in time the associated initial value problems, with moreover a minimal Sobolev regularity. We also explain the emergence and propagation of inertial waves.