Vortices in a Stochastic Parabolic Ginzburg-Landau Equation

TL;DR

This paper investigates the behavior of vortices in a stochastic Ginzburg-Landau equation with gradient-type noise, demonstrating the convergence of associated Jacobians to measures concentrated on vortex points.

Contribution

It characterizes the limit points of Jacobians in a stochastic Ginzburg-Landau model, revealing vortex formation and measure concentration phenomena.

Findings

Jacobians are tight on a measure space

Limit points are sums of delta measures with integer weights

Vortices correspond to singular points of the solution

Abstract

We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The singular set of the solution coincides with the points at which the delta measures are centered.