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We develop a functional framework suitable for the treatment of partialdifferential equations and variational problems on evolving families of Banachspaces. We propose a definition for the weak time derivative that does not relyon the availability of a Hilbertian structure and explore conditions underwhich spaces of weakly differentiable functions (with values in an evolvingBanach space) relate to classical Sobolev--Bochner spaces. An Aubin--Lionscompactness result is proved. We analyse concrete examples of function spacesover time-evolving spatial domains and hypersurfaces for which we explicitlyprovide the definition of the time derivative and verify isomorphism propertieswith the aforementioned Sobolev--Bochner spaces. We conclude with the proof ofwell posedness for a class of nonlinear monotone problems on an abstractevolving space (generalising the evolutionary $p$-Laplace equation on a movingdomain or surface) and identify some additional problems that can be formulatedwith the setting developed in this work.

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