Calculus of Variations and Geometric Measure Theory

A. Mielke - U. Stefanelli

Weighted energy-dissipation functionals for gradient flows

created by stefanell on 19 Mar 2009
modified on 26 Sep 2013


Published Paper

Inserted: 19 mar 2009
Last Updated: 26 sep 2013

Journal: ESAIM Control Optim. Calc. Var.,
Volume: 17
Number: 1
Pages: 52--85
Year: 2011


We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by {\sc Mielke & Ortiz} (2008). 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 {\sc Conti & Ortiz} (2008).

Keywords: Gradient Flow, Variational Principle, Convergence