Weighted energy-dissipation functionals for gradient flows

Alexander Mielke; Ulisse Stefanelli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 52-85
  • ISSN: 1292-8119

Abstract

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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.].

How to cite

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Mielke, Alexander, and Stefanelli, Ulisse. "Weighted energy-dissipation functionals for gradient flows." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 52-85. <http://eudml.org/doc/276333>.

@article{Mielke2011,
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.]. },
author = {Mielke, Alexander, Stefanelli, Ulisse},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Variational principle; gradient flow; convergence; abstract evolution; microstructure evolution},
language = {eng},
month = {2},
number = {1},
pages = {52-85},
publisher = {EDP Sciences},
title = {Weighted energy-dissipation functionals for gradient flows},
url = {http://eudml.org/doc/276333},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Mielke, Alexander
AU - Stefanelli, Ulisse
TI - Weighted energy-dissipation functionals for gradient flows
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 52
EP - 85
AB - 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.].
LA - eng
KW - Variational principle; gradient flow; convergence; abstract evolution; microstructure evolution
UR - http://eudml.org/doc/276333
ER -

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