Thin-film limit of the Ginzburg-Landau heat flow in a curved thin domain

TL;DR

This paper investigates the behavior of the Ginzburg-Landau heat flow in a curved thin domain as it collapses onto a surface, establishing convergence, limit equations, and difference estimates under various initial data conditions.

Contribution

It provides a rigorous analysis of the thin-film limit of the Ginzburg-Landau heat flow in curved geometries, including convergence results and explicit difference estimates.

Findings

Weak convergence of solutions to the limit surface

Derivation of the limit equation on the surface

Explicit difference estimates related to domain thickness

Abstract

We consider the Ginzburg-Landau heat flow without magnetic effect in a curved thin domain under the Naumann boundary condition. When the curved thin domain shrinks to a given closed hypersurface as the thickness of the thin domain tends to zero, we show that the weighted average of a weak solution to the thin-domain problem converges weakly on the limit surface under the assumption that the initial data is of class $L^\infty$ and satisfies some conditions. Moreover, under the same assumption, we derive a limit equation by characterizing the limit function as a weak solution, and prove a difference estimate on the limit surface of an averaged weak solution to the thin-domain problem and a weak solution to the limit problem explicitly in terms of the thickness of the thin domain. We also derive a difference estimate in the curved thin domain of weak solutions to the thin-domain problem and to the limit problem, but without requiring that the initial data of the thin-domain problem is of class $L^\infty$.