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: COCV 14 (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. Solids 56 (2008) 1885-1904.].
Mots clés : variational principle, gradient flow, convergence
@article{COCV_2011__17_1_52_0, author = {Mielke, Alexander and Stefanelli, Ulisse}, title = {Weighted energy-dissipation functionals for gradient flows}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {52--85}, publisher = {EDP-Sciences}, volume = {17}, number = {1}, year = {2011}, doi = {10.1051/cocv/2009043}, mrnumber = {2775186}, zbl = {1218.35007}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2009043/} }
TY - JOUR AU - Mielke, Alexander AU - Stefanelli, Ulisse TI - Weighted energy-dissipation functionals for gradient flows JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 52 EP - 85 VL - 17 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009043/ DO - 10.1051/cocv/2009043 LA - en ID - COCV_2011__17_1_52_0 ER -
%0 Journal Article %A Mielke, Alexander %A Stefanelli, Ulisse %T Weighted energy-dissipation functionals for gradient flows %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 52-85 %V 17 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009043/ %R 10.1051/cocv/2009043 %G en %F COCV_2011__17_1_52_0
Mielke, Alexander; Stefanelli, Ulisse. Weighted energy-dissipation functionals for gradient flows. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 52-85. doi : 10.1051/cocv/2009043. http://www.numdam.org/articles/10.1051/cocv/2009043/
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