Tadahisa Funaki (Waseda University, Japan)

Friday 7:30-8:30

Title: Motion by mean curvature from interacting particle systems

Abstract: The talk is on the derivation of motion by mean curvature (MMC) from two types of interacting particle systems via hydrodynamic limit. The hydrodynamic limit means, in probability group, a scaling limit in space and time for microscopic interacting systems with stochastic evolutional rule, which leads to nonlinear PDEs (not necessarily fluid mechanical equations) due to an averaging effect caused by the local ergodic property of the microscopic systems.

The microscopic systems we consider are Glauber-Zero range process and Glauber-Kawasaki dynamics with speed change. Glauber part controls creation and annihilation of particles, while the other part prescribes the interaction rule for particles performing random walks. For for zero range case, interaction occurs at the same site, while a hard core exclusion rule is introduced for Kawasaki case.

Microscopically, the system exhibits a phase separation to sparse and dense regions of particles and, macroscopically, the interface separating these two regions evolves under the MMC. We pass by the Allen-Cahn equation with nonlinear Laplacian at an intermediate level. The so-called Boltzmann-Gibbs principle plays a fundamental role. We also rely on Schauder estimate for quasilinear discrete PDEs.

The talk is based on joint works with S. Sethuraman, D. Hilhorst, P. El Kettani and H. Park (arXiv:2004.05276, arXiv:2112.13081), S. Sethuraman (arXiv:2112.13973), P. van Meurs, S. Sethuraman and K. Tsunoda (soon on arXiv).