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

2025/06/10 by Ken Abe, Abe, Ken, In‐Jee Jeong +5
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2506.08394

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity κ>0 and a force proportional to √κ on the flat d-torus \mathbbT^d for d≥ 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 H¹(\mathbbT^d) with law D(B)=μ as a non-resistive limit κ→ 0 of statistically stationary solutions B_κ(x,t). For d=2, the measure μ does not concentrate on any compact sets in H¹(\mathbbT²) with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium B(x) are almost surely not finite Fourier mode solutions.

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