Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
| Autor: | Amal Alphonse, Diogo Caetano, Ana Djurdjevac, Charles M. Elliott |
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| Rok vydání: | 2023 |
| Předmět: |
35K90
time derivatives moving domains 46G05 Applied Mathematics 500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik Parabolic PDEs function spaces Functional Analysis (math.FA) Mathematics - Functional Analysis Banach spaces Mathematics - Analysis of PDEs nonlinear PDEs FOS: Mathematics evolving surfaces 35R37 35R01 Analysis Analysis of PDEs (math.AP) |
| Zdroj: | Journal of Differential Equations. 353:268-338 |
| ISSN: | 0022-0396 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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