Abstract

 Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergence-free vector fields that (formally) preserves the knot structure of their integral lines. (See also the book by V.I. Arnold and B. Khesin.)
The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.-L. Lions for the Euler equations and Ambrosio-Gigli-Savar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.

Abstract