Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs

TL;DR

This paper introduces a new functional framework for PDEs on evolving Banach spaces, defining weak derivatives without Hilbertian assumptions, and proves compactness and well-posedness results for nonlinear problems on moving domains.

Contribution

It develops a novel approach to define weak derivatives in evolving Banach spaces and establishes key compactness and isomorphism results applicable to PDEs on moving domains.

Findings

Established a new weak derivative definition independent of Hilbert structure

Proved an Aubin--Lions type compactness theorem for evolving spaces

Demonstrated well-posedness for nonlinear PDEs on moving domains

Abstract

We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev--Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary $p$-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.