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

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