On some linear parabolic PDEs on moving hypersurfaces

TL;DR

This paper establishes existence and uniqueness results for various linear parabolic PDEs on moving hypersurfaces using an abstract framework on evolving Hilbert spaces, demonstrating its applicability to complex surface and bulk-surface problems.

Contribution

The paper extends an abstract framework for linear parabolic equations to moving hypersurfaces, enabling well-posedness analysis of surface, bulk, and coupled PDEs.

Findings

Proved well-posedness for surface heat equations on moving hypersurfaces.

Extended the framework to coupled bulk-surface systems.

Validated the approach for equations with dynamic boundary conditions.

Abstract

We consider existence and uniqueness for several examples of linear parabolic equations formulated on moving hypersurfaces. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. In order to prove the well-posedness, we make use of an abstract framework presented in a recent work by the authors which dealt with the formulation and well-posedness of linear parabolic equations on arbitrary evolving Hilbert spaces. Here, after recalling all of the necessary concepts and theorems, we show that the abstract framework can applied to the case of evolving (or moving) hypersurfaces, and then we demonstrate the utility of the framework to the aforementioned problems.