تقرير
Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
| العنوان: | Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs |
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| المؤلفون: | Alphonse, Amal, Caetano, Diogo, Djurdjevac, Ana, Elliott, Charles M. |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Functional Analysis |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1016/j.jde.2022.12.032 |
| URL الوصول: | http://arxiv.org/abs/2105.07908 |
| رقم الأكسشن: | edsarx.2105.07908 |
| قاعدة البيانات: | arXiv |
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