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 L^2 space. This choice is motivated by the nature of the operator that characterizes the turbulent noise in our model. The L^2 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. The last part of the work establishes a scaling limit theorem for the Stefan problem incorporating a mushy region on a torus, demonstrating that solutions to stochastic variants with turbulent transport terms converge to the solution to a deterministic partial differential equation. (joint work with Franco Flandoli and Dan Goreac)
A Stochastic Stefan Problem With Mushy Region and Turbulent Transport Noise
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