WIAS Preprint No. 2994, (2023)

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


  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • Caetano, Diogo
  • Djurdjevac, Ana
  • Elliott, Charles M.

2020 Mathematics Subject Classification

  • 35K90 46G05 35R37 35R01


  • Parabolic PDEs, function spaces, moving domains, evolving surfaces, nonlinear PDEs




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.

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