MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations

TL;DR

This paper studies the behavior of randomly forced resistive magnetic relaxation equations as resistivity approaches zero, establishing the existence of non-resistive magnetohydrostatic equilibria and analyzing their properties.

Contribution

It constructs non-resistive MHS equilibria as limits of stationary solutions and analyzes their measure-theoretic and Fourier properties.

Findings

Existence of path-wise global solutions and invariant measures for the system.

Construction of a non-resistive MHS equilibrium in the zero-resistivity limit.

In 2D, the equilibrium measure does not concentrate on finite Fourier modes.

Abstract

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity $\kappa>0$ and a force proportional to $\sqrt{\kappa}\ $ on the flat $d$-torus $\mathbb{T}^{d}$ for $d\geq 2$. We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium $B(x)$ in $H^{1}(\mathbb{T}^{d})$ with law $D(B)=\mu$ as a non-resistive limit $\kappa\to 0$ of statistically stationary solutions $B_{\kappa}(x,t)$. For $d=2$, the measure $\mu$ does not concentrate on any compact sets in $H^{1}(\mathbb{T}^{2})$ with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium $B(x)$ are almost surely not finite Fourier mode solutions.