arXiv subjects

Thomas Frenzel

Publications and source records attributed to Thomas Frenzel.

2 recordsLinked to original sources

arXiv2026

From diffusion to transmission via EDP-convergence: a paradigmatic multiscale limit

We consider nonlinear diffusion equations on an interval where the diffusion coefficient in a small region near the center is scaled such that it approximates a transmission condition at a membrane. While the limiting behavior of the solutions is well-understood, we study the convergence in the sense of the energy-dissipation principle (EDP) of associated gradient structures given in terms of a free energy and a dissipation potential. EDP-convergence provides a uniquely specified limiting gradient structure that reformulates the transmission condition in terms of an effective kinetic relation for the membrane, which relates the jump of the chemical potential and the flux through the membrane. We show how \AAA properties of the chosen free energy and the mobility of small-scale diffusion migrate to the effective kinetic relation. A surprising result is that starting from the linear Onsager relation of Otto's gradient structure for the linear diffusion equation, one obtains an exponentially growing kinetic relation, the so-called Marcelin-De Donder kinetic.

math.AP↗

arXiv2018

A gradient system with a wiggly energy and relaxed EDP-convergence

If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic effects. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. This new notion generalizes the concept of EDP-convergence, which was introduced in arXiv:1507.06322, and is called "relaxed EDP-convergence". Both notions are based on De Giorgi's energy-dissipation principle, however the special structure of the dissipation functional in terms of the primal and dual dissipation potential is, in general, not preserved under Gamma-convergence. By investigating the kinetic relation directly and using general forcings we still derive a unique macroscopic dissipation potential. The wiggly-energy model of James et al serves as a prototypical example where this nontrivial limit passage can be fully analyzed.

math.AP↗