We discuss how optimal transport, which is a theory for matching different distributions in a cost effective way, is applied to the supercooled Stefan problem, a free boundary problem that describes the interface dynamics of supercooled water freezing into ice. This problem exhibits a highly unstable behaviour and its mathematical study has been limited mostly to one space dimension, and widely open for multi-dimensional cases. We consider a version of optimal transport problem that considers stopping of the Brownian motion, whose solution is then translated into a solution to the supercooled Stefan problem in general dimensions.

In this talk, we consider the Boltzmann equation in general 3D toroidal domains with a specular reflection boundary condition. So far, it is a well-known open problem to obtain the low-regularity solution for the Boltzmann equation in general non-convex domains because there are grazing cases, such as inflection grazing. Thus, it is important to analyze trajectories which cause grazing. We will provide new analysis to handle these trajectories in general 3D toroidal domains.