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The Stefan problem with mushy region as a scaling limit of stochastic PDE with turbulent transport

Ioana Ciotir, Franco Flandoli, Dan Goreac

TL;DR

This work analyzes the scaling limit of a stochastic Stefan problem with a mushy region under turbulent transport. By formulating the system through maximal monotone operators $ \Psi $ and $ g $, it shows that as the noise intensity sequence $\{\alpha^N\} $ satisfies $ \|\alpha^N\|_{l^{\infty}} \to 0 $, the law of the stochastic solutions $ X^N $ converges to the Dirac mass at the unique solution of the deterministic equation $ dX - \Delta\Psi(X)\,dt - \Delta g(X)\,dt = F X(0,\xi) = x $. The stochastic term vanishes in the limit, while the diffusion is augmented by $ -\Delta g(X) $, reflecting turbulence-enhanced melting. The proof uses energy estimates, tightness via compactness, and Skorokhod representation to identify the limit as the solution to the deterministic porous-media-type PDE, relying on the monotone-operator framework to ensure uniqueness. This provides a rigorous link between turbulent transport in phase-change problems and deterministic mushy-region evolution, with implications for ice-melting acceleration due to turbulence.

Abstract

This work establishes a scaling limit theorem for the Stefan problem incorporating a mushy region, demonstrating that solutions to stochastic variants with turbulent transport terms converge to the solution to a deterministic partial differential equation. The analysis builds upon recent advances in stochastic phase-change modeling and turbulent flow mathematics in [5]. In the physical interpretation of an ice melting process, our result shows that turbulence accelerates ice melting.

The Stefan problem with mushy region as a scaling limit of stochastic PDE with turbulent transport

TL;DR

This work analyzes the scaling limit of a stochastic Stefan problem with a mushy region under turbulent transport. By formulating the system through maximal monotone operators and , it shows that as the noise intensity sequence satisfies , the law of the stochastic solutions converges to the Dirac mass at the unique solution of the deterministic equation . The stochastic term vanishes in the limit, while the diffusion is augmented by , reflecting turbulence-enhanced melting. The proof uses energy estimates, tightness via compactness, and Skorokhod representation to identify the limit as the solution to the deterministic porous-media-type PDE, relying on the monotone-operator framework to ensure uniqueness. This provides a rigorous link between turbulent transport in phase-change problems and deterministic mushy-region evolution, with implications for ice-melting acceleration due to turbulence.

Abstract

This work establishes a scaling limit theorem for the Stefan problem incorporating a mushy region, demonstrating that solutions to stochastic variants with turbulent transport terms converge to the solution to a deterministic partial differential equation. The analysis builds upon recent advances in stochastic phase-change modeling and turbulent flow mathematics in [5]. In the physical interpretation of an ice melting process, our result shows that turbulence accelerates ice melting.
Paper Structure (5 sections, 1 theorem, 53 equations)

This paper contains 5 sections, 1 theorem, 53 equations.

Table of Contents

  1. Introduction
  2. The physical model
  3. Functional setting
  4. The mathematical model
  5. The scaling limit and the main result

Key Result

Theorem 2

Let $\left\{ \alpha ^{N}\right\} _{N\in \mathbb{N}}\subseteq l^{2}\left( \mathbb{Z}_{0}^{2}\right)$ be a sequence satisfying the assumptions (ip1) and (ip2). We denote Then, the family $\left\{ \nu ^{N}\right\}$ is tight on $C\left( \left[ 0,T\right] ;H^{-1}\left( \Pi ^{2}\right) \right)$, and it converges weakly to the Dirac measure $\delta _{X}$ where $X$ is the unique solution of the equation

Theorems & Definitions (3)

  • Definition 1
  • Theorem 2
  • Remark 3