Well-posedness of the heat equation in domains with topological transitions

TL;DR

This paper establishes the well-posedness of the heat equation in evolving domains with topological changes, using novel anisotropic space-time function spaces and Morse theory to handle complex domain transitions.

Contribution

It introduces anisotropic space-time function spaces for domains with topological transitions and proves existence, uniqueness, and estimates for solutions in this setting.

Findings

Framework applies to domains with splitting and merging.

Functional-analytic properties of new spaces are established.

Existence and uniqueness of solutions are proven.

Abstract

We analyze a linear parabolic equation with homogeneous Dirichlet boundary conditions posed in domains whose evolution may involve topological transitions. The domains are described as sublevel sets of a smooth space-time level set function, allowing for transitions such as domain splitting and merging and the creation or vanishing of islands and holes. We introduce anisotropic space-time function spaces that extend the classical Bochner spaces used in cylindrical domains and establish key functional-analytic properties of these spaces, including the density of compactly supported smooth functions. This framework enables the application of the Babu\v{s}ka-Banach theorem, yielding existence, uniqueness, and a priori estimates for weak solutions. The analysis applies to domain evolutions generated by level set functions with isolated nondegenerate critical points, which correspond to the generic topology changes classified by Morse theory in two and three spatial dimensions.