Stefan Problems for Reflected SPDEs Driven by Space-Time White Noise

TL;DR

This paper establishes existence and uniqueness results for a one-dimensional Stefan problem involving reflected stochastic partial differential equations driven by space-time white noise, incorporating reflection measures and boundary conditions.

Contribution

It introduces a novel analysis of Stefan problems with space-time white noise, reflection measures, and boundary conditions, extending classical models to stochastic and reflected settings.

Findings

Solutions exist until almost surely positive blow-up times

Numerical simulations suggest the boundary condition is nearly necessary

The model captures phase evolution with stochastic interface dynamics

Abstract

We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space-time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary.