An abstract framework for parabolic PDEs on evolving spaces

We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formul...

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Bibliographic Details
Published in:arXiv.org 2015-07
Main Authors: Amal Alphonse, Elliott, Charles M, Stinner, Björn
Format: Article
Language:English
Subjects:
Domains
Evolution
Formulations
Hilbert space
Hyperspaces
Operators (mathematics)
Parabolic differential equations
Partial differential equations
Well posed problems
Online Access:Get full text
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Summary:We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.
ISSN:2331-8422
DOI:10.48550/arxiv.1403.4500