On evolutionary Gamma convergence for gradient systems
- Mielke, Alexander
2010 Mathematics Subject Classification
- 35B27 35K55 47H20 49S05 49J40 49J45
- Variational evolution, energy functional, dissipation potential, dissipation distance, gradient flows, Gamma convergence, Mosco convergence, well-prepared initial conditions, rate-independent systems, abstract chain-rule, energy-dissipation balance, integrated evolutionary variational estimate, energetic solutions
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 Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions.
We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results.
- A. Mielke, Chapter 3: On Evolutionary $Gamma$-Convergence for Gradient Systems, in: Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, A. Muntean, J.D.M. Rademacher, A. Zagaris, eds., vol. 3 of Lecture Notes in Applied Mathematics and Mechanics, Springer International Publishing, Heidelberg et al., 2016, pp. 187--249