Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation

TL;DR

This paper introduces a topology-preserving diffusion model for divergence-free vector fields, addressing limitations of the heat equation in maintaining the integral line structure of such fields.

Contribution

It proposes a novel concept of dissipative solutions for these diffusion equations, bridging ideas from fluid mechanics and metric space analysis.

Findings

The diffusion equations preserve the topology of divergence-free vector fields.

Dissipative solutions share features with solutions to Euler equations and curves of maximal slopes.

The approach extends the understanding of topology-preserving diffusion in fluid mechanics.

Abstract

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, on can find examples of topology-preserving diffusion equations for divergence-free vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of "dissipative solutions", which shares common features with both P.-L. Lions' dissipative solutions to the Euler equations and the concept of "curves of maximal slopes", a la De Giorgi, recently used to study the scalar heat equation in very general metric spaces.