A Generalization of the Aubin-Lions-Simon Compactness Lemma for Problems on Moving Domains

TL;DR

This paper extends the Aubin-Lions-Simon compactness lemma to variable Hilbert spaces on moving domains, enabling the analysis of nonlinear PDEs with evolving boundaries, such as fluid-structure interaction problems.

Contribution

It introduces a generalized compactness theorem for moving domain problems, providing conditions on domain regularity and applications to Navier-Stokes and fluid-structure interaction models.

Findings

Extended Aubin-Lions-Simon lemma for moving domains

Applicable to Navier-Stokes equations on non-cylindrical domains

Facilitates existence proofs for nonlinear boundary problems

Abstract

This work addresses an extension of the Aubin-Lions-Simon compactness result to generalized Bochner spaces $L^2(0,T;H(t))$, where $H(t)$ is a family of Hilbert spaces, parameterized by $t$. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin-Lions-Simon compactness result holds. Concrete examples of the application of the compactness theorem are presented, including a classical problem for the incompressible, Navier-Stokes equations defined on a {\sl given} non-cylindrical domain, and a class of fluid-structure interaction problems for the incompressible, Navier-Stokes equations, coupled to the elastodynamics of a Koiter shell. The compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear, moving boundary problems, using Rothe's method.