Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials
Mathematisch-Naturwissenschaftliche Fakultät
We consider a non-negative and one-homogeneous energy functional J on a Hilbert space.
The paper provides an exact relation between the solutions of the associated gradient-flow
equations and the energetic solutions generated via the rate-independent system given in terms
of the time-dependent functional E(t, u) = tJ(u) and the norm as a dissipation distance. The
relation between the two flows is given via a solution-dependent reparametrization of time
that can be guessed from the homogeneities of energy and dissipations in the two equations.
We provide several examples including the total-variation flow and show that equivalence
of the two systems through a solution dependent reparametrization of the time. Making
the relation mathematically rigorous includes a careful analysis of the jumps in energetic
solutions which correspond to constant-speed intervals for the solutions of the gradient-flow
equation. As a major result we obtain a non-trivial existence and uniqueness result for the
energetic rate-independent system.
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