Alphonse, Amal and Caetano, Diogo and 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.

Full text not available from this repository.

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: | Article |
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Subjects: | Mathematical and Computer Sciences Mathematical and Computer Sciences > Mathematics Mathematical and Computer Sciences > Mathematics > Applied Mathematics |

Divisions: | Department of Mathematics and Computer Science > Institute of Mathematics |

ID Code: | 2942 |

Deposited By: | Monika Drueck |

Deposited On: | 18 Apr 2023 08:20 |

Last Modified: | 18 Apr 2023 08:20 |

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