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Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
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Alphonse, Amal, Caetano, Diogo, Djurdjevac, Ana and Elliott, Charles M. (2023) Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs. Journal of Differential Equations, 353 . pp. 268-338. doi:10.1016/j.jde.2022.12.032 ISSN 0022-0396.
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Official URL: https://doi.org/10.1016/j.jde.2022.12.032
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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
SWORD Depositor: | Library Publications Router | ||||||||
Library of Congress Subject Headings (LCSH): | Function spaces, Differential equations, Elliptic, Banach spaces, Differential equations, Partial | ||||||||
Journal or Publication Title: | Journal of Differential Equations | ||||||||
Publisher: | Elsevier | ||||||||
ISSN: | 0022-0396 | ||||||||
Official Date: | 25 April 2023 | ||||||||
Dates: |
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Volume: | 353 | ||||||||
Page Range: | pp. 268-338 | ||||||||
DOI: | 10.1016/j.jde.2022.12.032 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 1 February 2023 | ||||||||
Date of first compliant Open Access: | 3 February 2023 |
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