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

TL;DR

This paper investigates the long-term behavior of Moffatt's magnetic relaxation equation with Darcy regularization, aiming to prove convergence to Euler equilibria in 2D and 2.5D flows within various domains.

Contribution

It provides new proofs of convergence for specific classes of equilibria in 2D and 2.5D flows, extending understanding of magnetic relaxation dynamics.

Findings

Convergence proven for non-constant shear flows in 2D channels.

Geometric approach used to address 2.5D equilibria.

Results support the conjecture of convergence to Euler equilibria.

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 $\Omega\times \mathbb{R}$, where $\Omega\subset \mathbb{R}^2$ can be a periodic channel or any bounded domain.