Stochastic Modeling of the Behavior of Phase Boundaries in the Mean Curvature Flow and Simulation Analyses

Abstract

This paper is concerned with behavior analyses of the stochastic mean curvature flows of some phase boundaries. The phase boundary is defined as a curve or a surface separating different physical states such as a water-ice interface. The mean curvature flow is motion that the phase boundary moves with a normal velocity equals the mean curvature at each point on the phase boundary. We formulate the mean curvature flow by a stochastic level set equation, which is a nonlinear stochastic partial differential equation with a unique solution in a sense of a stochastic viscosity solution. In numerical simulations, sample behaviors of the stochastic mean curvature flows of barbell and torus shapes are studied. And we numerically show that sample behaviors of the phase boundary in stochastic mean curvature flows have a possibility of changing topologically by the random noise without any fattening phenomena. And in the case where the fattening occurs, we show that the random noise plays a role of selecting a unique solution among the set of possible solutions.