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arXiv · 2503.02314

Stochastic Stefan problem on moving hypersurfaces: an approach by a new framework of nonhomogeneous monotonicity

Tianyi Pan·Wei Wang·Jianliang Zhai·Tusheng Zhang
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Abstract

The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a fixed hypersurface with a new kind of nonhomogeneous monotonicity involving a family of time-dependent operators. This new class of SPDEs is of independent interest and can also be applied to solve many other interesting models such as the stochastic $p$-Laplacian equations, stochastic Allen-Cahn equation and stochastic heat equations on time-dependent domains or hypersurfaces. (Monotone) Operator-valued calculus and geometric analysis of moving hypersurfaces play important roles in the study. Moreover, a forthcoming result on the well-posedness of stochastic 2D Navier-Stokes equation on moving domains is also based on our framework.

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Tianyi Pan, Wei Wang, Jianliang Zhai, Tusheng Zhang. 2025-03-04. Stochastic Stefan problem on moving hypersurfaces: an approach by a new framework of nonhomogeneous monotonicity. https://arxiv.org/abs/2503.02314

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