WED approach to dissipative problems and applications
Stefano Melchionna, University of Vienna
Sala conferenze IMATI-CNR - Thursday, November 26, 2015 h.15:00
Abstract. Given a target dissipative problem (P), the so-called Weighted Energy-Dissipation (WED) procedure consists in defi ning a global parameter dependent functional I_epsilon over trajectories and proving
that its minimizers converge up to subsequences to solutions to the target system (P) as the parameter epsilon goes to 0. Such a global-in-time variational approach to evolution problems is interesting,
since it paves the way to the application of tools and techniques of the calculus of variation (e.g.Direct Method, relaxation, Gamma-convergence) in the evolutive setting. This brings also a new tool for
checking qualitative properties of solutions and comparison principles. Moreover, the minimization
problem features, typically, more regular solutions, since the Euler-Lagrange system associated with
the minimization of I_epsilon corresponds to an elliptic-in-time regularisation (P)^epsilon of the original problem (P).
By combining the minimization of a WED type functional and a fixed point argument, we prove
existence of solutions to an elliptic-in-time regularization of nonpotential perturbations of gradient
flows and convergence for epsilon that goes to 0. We also apply the WED formalism to check qualitative properties of solutions and comparison principles for some classes of dissipative equations.
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