Winter School 2014

Calculus of Variations in Physics and Materials Science (February 10 - 14, 2014)

Topics

Variational Models in Fracture Mechanics
Gianni Dal Maso, SISSA, Trieste, Italy
Abstract: The course will present an overview on the variational approach to quasistatic evolution problems in fracture mechanics, in the more general framework of the theory of rate independent processes. The course will analyse different models and discuss their mechanical implications and the mathematical problems that they introduce. The common feature of all these models is that the solutions are obtained through approximation by time discretization and that the approximate solutions are obtained by solving suitable incremental minimum problems. The continuous time solutions are then obtained by passing to the limit as the time step tends to zero. The course will introduce the function spaces used to construct a correct mathematical formulation of these models and will present the main properties used in the proofs of the results. The course will also cover the most recent results on brittle cracks in hyperelastic bodies, in the context of finite elasticity.
Stochastic Origins of Energies and Gradient Flows: a Modeling Guide
Mark Peletier, TU Eindhoven, The Netherlands
Abstract: In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In this series of talks I will build an understanding of the modelling arguments that underlie the use of energies, entropies, and gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. I will explain all these concepts in detail in the lectures. I will assume that the participants have a basic understanding of measure theory, Sobolev spaces, and some of the more common types of partial differential equations. No prior knowledge of optimal transport, Wasserstein gradient flows, or probability is required.
Energy, Energy-Dissipation, and Distance in Gradient Flow Systems
Maria G. Westdickenberg, RWTH Aachen, Germany
Abstract: A gradient flow system has a naturally occuring triple of energy, energy-dissipation, and induced distance. It is well known that if the energy is nondegenerate in the neighborhood of a local minimum point, then the corresponding gradient flow solution with initial data in a small neighborhood of this point converges in time to the minimum. In these lectures we explore other aspects of the energy landscape and gradient flow dynamics that can be explored via the relationships among the nonlinear quantities in the natural triple named above. In particular, we offer sufficient conditions for a gradient flow system to demonstrate so-called dynamic metastability. We apply this framework to give a simple proof of metastability of the 1-d Allen-Cahn equation. We then explain how a mismatch in scales prevents a direct application of the same method to the 1-d Cahn-Hilliard equation. We resolve this difficulty via a new framework for establishing convergence rates in time for gradient flows with respect to mildly nonconvex energies. This work is joint with Felix Otto.

Invited Speakers

Gianni Dal Maso, SISSA, Trieste, Italy
Variational Models in Fracture Mechanics
Mark Peletier, TU Eindhoven, The Netherlands
Stochastic Origins of Energies and Gradient Flows: a Modeling Guide
Maria G. Westdickenberg, RWTH Aachen, Germany
Energy, Energy-Dissipation, and Distance in Gradient Flow Systems