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Weighted energy-dissipation functionals for gradient flows

Published online by Cambridge University Press:  30 October 2009

Alexander Mielke  and
Ulisse Stefanelli
Alexander Mielke
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany. [email protected] Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany.
Ulisse Stefanelli
Affiliation:
IMATI – CNR, v. Ferrata 1, 27100 Pavia, Italy. [email protected]
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Abstract

We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz [ESAIM: COCV14 (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from [S. Conti and M. Ortiz, J. Mech. Phys. Solids56 (2008) 1885–1904.].

Keywords

Variational principlegradient flowconvergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 52 - 85
Copyright
© EDP Sciences, SMAI, 2009

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