On Moffatt's magnetic relaxation for 2D and 2.5D flows

TL;DR

The article analyzes Moffatt's magnetic relaxation equation with Darcy-type regularization, proving relaxation to steady Euler states for two settings: (i) a class of non-constant shear flows in a 2D periodic channel, and (ii) 2.5D equilibria on domains $\Omega\times\mathbb{R}$ using a geometric approach. In the 2D channel, the authors decompose the perturbation into a mean part $a(x_2,t)$ and a transverse part $f$, derive coupled evolution equations, and establish exponential decay of $f$ via semigroup estimates for a linear operator $L_a$, with smallness of $a$ ensuring global relaxation. The 2.5D analysis couples the horizontal MRE to a scalar height field $g$ through $u_3=B_H\cdot\nabla_H g$ and shows that, for shear-type horizontal fields with nonvanishing components, $g$ relaxes to a steady state and the horizontal field approaches a steady Euler solution. A geometric framework further extends the relaxation theory to general fields $B$ whose level sets are periodic orbits, yielding exponential relaxation in $L^2$ and, under bounded orbit periods, in certain Sobolev norms; the framework leverages the coarea formula and orbit-wise heat flow. Overall, the work advances understanding of topology-preserving magnetic relaxation by linking linear semigroup decay, nonlinear bootstrap, and geometric damping mechanisms across 2D and 2.5D settings, with implications for converging to steady MHD/Euler states in bounded domains.

Abstract

We study the Moffatt's magnetic relaxation equation with Darcy-type regularization for the constitutive law. This is a topology-preserving dissipative equation, whose solutions are conjectured to converge in the infinite time limit towards equilibria of the incompressible Euler equations. Our goal is to prove this conjectured property for various equilibria in various domains. The first result concerns a class of non-constant shear flows in a 2D periodic channel. In the second result, by adopting a geometric approach, we address a class of 2.5D equilibria in $Ω\times \mathbb{R}$, where $Ω\subset \mathbb{R}^2$ can be a periodic channel or any bounded domain.

Theorems & Definitions (66)

• Lemma 1.1: $L^2$ Estimate for General Domain
• proof
• Theorem 1.2: Informal version
• Definition 2.1: Leray Projection
• Definition 2.2
• Theorem 2.3
• Lemma 2.4: Commutator Estimate
• proof
• Lemma 2.5: Commutator estimate for Leray's projector
• proof