2020/10/01 by Alexander Mielke, Mielke, Alexander
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2010.00314
openalex publication_date 2020/10/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We consider a non-negative and one-homogeneous energy functional mathcal J\non a Hilbert space. The paper provides an exact relation between the solutions\nof the associated gradient-flow equations and the energetic solutions generated\nvia the rate-inpendent system given in terms of the time-dependent functional\n mathcal E(t,u)=t mathcal J(u) and the norm as a dissipation distance. The\nrelation between the two flows is given via a solution-dependent\nreparametrization of time that can be guessed from the homogeneities of energy\nand dissipations in the two equations. We provide several examples including\nthe total-variation flow and show that equivalence of the two systems through a\nsolution dependent reparametrization of the time. Making the relation\nmathematically rigorous includes a careful analysis of the jumps in energetic\nsolutions which correspond to constant-speed intervals for the solutins of the\ngradient-flow equation. As a major result we obtain a non-trivial existence and\nuniqueness result for the energetic rate-independent system.\n