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

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.

References in corpus (6)

• Evolving surface finite element method for the Cahn-Hilliard equation
• An abstract framework for parabolic PDEs on evolving spaces
• Fluid-structure interaction modeling of blood flow in the pulmonary arteries using the unified continuum and variational multiscale formulation
• Reflections on Dubinskii's nonlinear compact embedding theorem
• A Stefan problem on an evolving surface
• Sobolev space of functions valued in a monotone Banach family

Cited by in corpus (4)

• A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential
• An evolving space framework for Oseen equations on a moving domain
• Existence of variational solutions to doubly nonlinear systems in general noncylindrical domains
• Decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces