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Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise

Ľubomír Baňas, Jean Daniel Mukam

TL;DR

This work rigorously justifies the sharp-interface limit for the stochastic Cahn–Hilliard equation with additive space–time white noise, proving convergence to the Mullins–Sekerka/Hele–Shaw problem in dimensions $d=2,3$ under sufficiently large noise scaling $σ$ and providing optimal convergence rates in fractional Sobolev and $L^p$ norms with $p\in(2,4]$. The analysis introduces a translated error framework, leveraging stochastic convolution, a random PDE for the error, and a stopping-time argument to control nonlinearities despite the low regularity of the noise; it also identifies minimal noise regularity needed for $H^1$-type convergence and delivers improved deterministic sharp-interface estimates. In the case of more regular noise, the authors show $\mathbb{H}^1$ convergence in $d=3$, and they extend the deterministic theory with sharper $L^p$-type bounds ($p\in(2,4]$) compared to previous results. Overall, the paper advances the mathematical understanding of stochastic diffuse-interface limits and provides tools and thresholds valuable for analysis and numerical approximation of stochastic phase-field models.

Abstract

We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise $ε^σ\dot{W}$ where $ε>0$ is an interfacial width parameter. We prove that, for sufficiently large scaling constant $σ>0$, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for $ε\rightarrow 0$. The convergence is shown in suitable fractional Sobolev norms as well as in the $L^p$-norm for $p\in (2, 4]$ in spatial dimension $d=2,3$. This generalizes the existing result for the space-time white noise to dimension $d=3$ and improves the existing results for smooth noise, which were so far limited to $p\in \left(2, \frac{2d+8}{d+2}\right]$ in spatial dimension $d=2,3$. As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the $\mathbb{H}^1$-norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.

Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise

TL;DR

This work rigorously justifies the sharp-interface limit for the stochastic Cahn–Hilliard equation with additive space–time white noise, proving convergence to the Mullins–Sekerka/Hele–Shaw problem in dimensions under sufficiently large noise scaling and providing optimal convergence rates in fractional Sobolev and norms with . The analysis introduces a translated error framework, leveraging stochastic convolution, a random PDE for the error, and a stopping-time argument to control nonlinearities despite the low regularity of the noise; it also identifies minimal noise regularity needed for -type convergence and delivers improved deterministic sharp-interface estimates. In the case of more regular noise, the authors show convergence in , and they extend the deterministic theory with sharper -type bounds () compared to previous results. Overall, the paper advances the mathematical understanding of stochastic diffuse-interface limits and provides tools and thresholds valuable for analysis and numerical approximation of stochastic phase-field models.

Abstract

We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise where is an interfacial width parameter. We prove that, for sufficiently large scaling constant , the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for . The convergence is shown in suitable fractional Sobolev norms as well as in the -norm for in spatial dimension . This generalizes the existing result for the space-time white noise to dimension and improves the existing results for smooth noise, which were so far limited to in spatial dimension . As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the -norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.
Paper Structure (6 sections, 13 theorems, 111 equations)

This paper contains 6 sections, 13 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Existence and regularity of the solution
  4. Sharp interface limit of the stochastic problem
  5. Limiting case with $\mathbb{H}^1$ spatial regularity
  6. The deterministic problem

Key Result

Proposition 2.1

Let $u^{\epsilon}_{\mathsf{A}}$ be the approximation in model5. Then for all $w\in \mathbb{H}^1$ with $\int_{\mathcal{D}}wdx=0$, the following holds where $C_0\geq 0$ is a constant independent of $w$ and $\epsilon$.

Theorems & Definitions (27)

  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 17 more