Boundary integral operators for the heat equation in time-dependent domains

TL;DR

This paper develops a functional analytical framework for boundary integral equations of the heat equation in time-dependent, non-cylindrical domains, addressing the complexities introduced by moving boundaries and their impact on layer operators.

Contribution

It introduces Sobolev spaces, trace lemmata, and mapping properties of layer operators for heat equations in non-cylindrical domains, extending previous work to time-dependent geometries.

Findings

Established Sobolev space framework for non-cylindrical domains

Derived mapping properties of layer operators in time-dependent settings

Analyzed the Neumann trace correction for moving boundaries

Abstract

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the $C^2$-diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators by mimicking the proofs of [M. Costabel, Boundary integral operators for the heat equation, Integral Equations and Operator Theory, 13(4):498-552, 1990]. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.