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

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 $L^\infty$.