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

Tianyi Pan, Wei Wang, Jianliang Zhai, Tusheng Zhang

TL;DR

The paper tackles the stochastic Stefan problem on moving hypersurfaces by mapping it to a transformed SPDE on a fixed domain with a time-dependent inner product and a novel nonhomogeneous monotonicity framework. It develops an operator calculus around a family of inner-product maps $\iota_t^*$ and a drift operator $\Phi(t)$, and proves well-posedness via a time-dependent Galerkin scheme plus a tailored Itô formula for $|\cdot|_t^2$, obtaining a robust equivalence between moving-surface and fixed-domain formulations. The results extend to stochastic diffusion-type equations on time-dependent domains and give an Itô-type energy identity for the $\dot{H}^{-1}$ norm under evolving geometry. This framework provides a versatile toolkit for SPDEs on moving manifolds and yields precise transport formulas that can aid in future analyses of stochastic Navier–Stokes and related systems on moving domains. Overall, the work advances the mathematical understanding of stochastic PDEs in evolving geometric settings and broadens applicability to a range of time-dependent problems.

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.

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

TL;DR

The paper tackles the stochastic Stefan problem on moving hypersurfaces by mapping it to a transformed SPDE on a fixed domain with a time-dependent inner product and a novel nonhomogeneous monotonicity framework. It develops an operator calculus around a family of inner-product maps and a drift operator , and proves well-posedness via a time-dependent Galerkin scheme plus a tailored Itô formula for , obtaining a robust equivalence between moving-surface and fixed-domain formulations. The results extend to stochastic diffusion-type equations on time-dependent domains and give an Itô-type energy identity for the norm under evolving geometry. This framework provides a versatile toolkit for SPDEs on moving manifolds and yields precise transport formulas that can aid in future analyses of stochastic Navier–Stokes and related systems on moving domains. Overall, the work advances the mathematical understanding of stochastic PDEs in evolving geometric settings and broadens applicability to a range of time-dependent problems.

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 -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.

Paper Structure

This paper contains 15 sections, 14 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. SPDEs with non-homogeneous monotonicity
  3. A motivating example
  4. Assumptions on the inner products
  5. SPDEs with nonhomogeneous monotonicity
  6. Proof of Theorem \ref{['wp']}
  7. Stochastic Stefan type problem on a moving hypersurface
  8. Preliminaries on moving hypersurfaces
  9. The geometry of moving hypersurfaces
  10. Main results of Section 3
  11. Verification of (\ref{['C1']})-(\ref{['C4']})
  12. Equivalence of solutions
  13. Verification of (\ref{['H1']})-(\ref{['H5']})
  14. Stochastic Transport formula for $\dot{H}^{-1}(\Gamma_\cdot)$ norm
  15. A variational formulation of $\iota_{-t}^*$

Key Result

Lemma 2.3

For the operators $\iota_t^*$ and $\Phi(t)$ on $H$ defined above, we have (i) For any $t\in[0,T]$, $x\in H$, $\iota_t^*x=x+\int_{0}^{t}\Phi(s)xds$. (ii) For any $s<t$ in $[0,T]$, $\|\iota_t^*-\iota_s^*\|_{\mathcal{L}(H)}\leq \int_{s}^{t}\|\Phi(r)\|_{\mathcal{L}(H)}dr$. (iii) For any $t\in[0,T]$, $\i

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 13 more