Gradient Flows and Rate-independent Evolutions: a Variational Approach
نویسنده
چکیده
The course will give a brief overview of the variational theory for gradient flows and rate-independent evolutions, trying to focus on the most important aspects: variational approximations, convergence results, energy-dissipation inequalities and metric characterization for gradient flows; energetic descriptions, BV solutions and optimal jump transitions, viscous approximations in the case of rate-independent evolutions. Some useful tools of convex and metric analysis will also be recalled. Here is a more detailed description of the topics and a few references.
منابع مشابه
Variational convergence of gradient flows and rate-independent evolutions in metric spaces
We study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, a...
متن کاملQuasi-static Evolutions in Linear Perfect Plasticity as a Variational Limit of Finite Plasticity: a One-dimensional Case
In the framework of the energetic approach to rate independent evolutions, we show that one-dimensional linear perfect plasticity can be obtained by linearization as a variational limit of a finite plasticity model with hardening proposed by A. Mielke (SIAM J. Math. Anal., 2004).
متن کاملA Variational Principle for Gradient Flows of Nonconvex Energies
We present a variational approach to gradient flows of energies of the form E = φ1−φ2 where φ1, φ2 are convex functionals on a Hilbert space. A global parameter-dependent functional over trajectories is proved to admit minimizers. These minimizers converge up to subsequences to gradient-flow trajectories as the parameter tends to zero. These results apply in particular to the case of non λ-conv...
متن کاملA Gradient Descent Procedure for Variational Dynamic Surface Problems with Constraints
Many problems in image analysis and computer vision involving boundaries and regions can be cast in a variational formulation. This means that m-surfaces, e.g. curves and surfaces, are determined as minimizers of functionals using e.g. the variational level set method. In this paper we consider such variational problems with constraints given by functionals. We use the geometric interpretation ...
متن کاملModeling solutions with jumps for rate-independent systems on metric spaces
Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generaliz...
متن کامل