Well-posedness for mean-field evolutions arising in superconductivity

TL;DR

This paper proves the existence, uniqueness, and regularity of solutions for a new class of mean-field equations modeling supercurrent density in type-II superconductors, connecting to fluid dynamics and shallow water equations.

Contribution

It introduces a novel family of fluid-like equations for supercurrent evolution, establishing global solutions and extending results to rough initial data.

Findings

Existence of global solutions for the new equations.

Uniqueness and regularity properties analyzed.

Connection to lake equations in fluid dynamics.

Abstract

We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in a joint work with Serfaty in certain regimes as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.