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Title: On evolutionary [Gamma]-convergence for gradient systems : in memory of Eduard, Waldemar, and Elli Mielke
Authors: Mielke, Alexander
Issue Date: 2014
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 1915, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional E and the dissipation potential R or the associated dissipation distance. We assume that the functionals depend on a small parameter and that the associated gradient systems have solutions u. We investigate the question under which conditions the limits u of (subsequences of) the solutions u are solutions of the gradient system generated by the [Gamma]-limits E0 and R0. Here the choice of the right topology will be crucial awell as additional structural conditions. We cover classical gradient systems, where R is quadratic, and rate-independent systems as well as the passage from classical gradient to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.
Keywords: Variational evolution; energy functional; dissipation potential; dissipation distance; gradient flows; Gamma convergence; well-prepared initial conditions; rate-independent systems; abstract chain-rule; enerty-dissipation balance integrated evolutionary variational estimate; energetic solutions
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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Appears in Collections:Mathematik



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