An Existence Result for a Stochastic Stefan Problem With Mushy Region and Turbulent Transport Noise
Ioana Ciotir, Franco Flandoli, Dan Goreac
TL;DR
This work proves the existence of a martingale solution for a stochastic Stefan problem featuring a mushy region and turbulent transport noise. By recasting the problem in an $L^2$ framework and mollifying the nonlinearities to obtain a strictly monotone operator, the authors bypass the standard $H^{-1}$ setting typically used for porous media-type equations. They rigorously construct the noise via a corpus of divergence-free modes, transform Stratonovich noise to Itô form with a correction term, and implement a Galerkin approximation together with tightness and Skorohod representation to pass to the limit. The result advances the mathematical understanding of stochastic phase-change problems with nonlinear transport effects and provides a robust existence theory for physically motivated turbulence models.
Abstract
This work is devoted to the proof of the existence of a martingale solution for a complex version of the stochastic Stefan problem. This particular formulation incorporates two important features: a mushy region and turbulent transport within the liquid phase. While our approach bears similarities to porous media equations, it differs in a crucial aspect. Instead of using the typical framework for such equations, we have chosen to work within an L2 space. This choice is motivated by the nature of the operator that characterizes the turbulent noise in our model. The L2 space provides a more natural and appropriate setting for handling this specific operator, allowing us to better capture and analyze the turbulent transport phenomena in the liquid phase of the Stefan problem.