Mathematics Colloquium
Sergey Nadtochiy
Illinois Institute of Technology
Title: Probabilistic solutions to Stefan equations
Date: Friday, March 31, 2023
Place and Time: LOV 101, 3:05-3:55 pm
Abstract.
This talk is concerned with the probabilistic methods for solving Stefan free-boundary PDEs (a.k.a. laplacian growth models). The latter equations appear in many models of fundamental physical and biological processes, such as: phase transition (i.e., melting/freezing), phase segregation (e.g., aging of alloys), crystal growth, neurons interaction, etc.. Despite their importance, to date, there exists no general existence and uniqueness theory for such equations due to the potential singularity of the solutions. Recently, the probabilistic methods, based on the analysis of associated mean-field particle systems and McKean-Vlasov equations, were successfully used to tackle the mathematical challenges that could not be addressed by the classical analytic methods, yielding new well-posedness results for certain types of Stefan equations. I will present an overview of the existing results, including the recently obtained proof of well-posedness of Stefan equation with surface tension under radial symmetry. This talk is based on joint works with F. Delarue, M. Shkolnikov, X. Zhang and Y. Guo.