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The stochastic nonlocal Cahn-Hilliard equation with regular potential and multiplicative noise

Andrea Di Primio, Christoph Hurm

TL;DR

The paper rigorously analyzes the stochastic nonlocal Cahn–Hilliard equation with a regular potential under multiplicative noise on bounded domains, proving existence of martingale solutions and, under Lipschitz diffusion, pathwise uniqueness and the existence of a probabilistically-strong solution. It develops a two-stage approximation (Yosida regularization plus Galerkin discretization) and employs stochastic compactness to pass to the limit, first in the Galerkin dimension and then in the regularization parameter, carefully identifying nonlinearities and stochastic integrals. The work also establishes nonlocal-to-local convergence, deriving rates under higher regularity and showing convergence without extra regularity via ε-uniform estimates and Gyöngy–Krylov methods. Altogether, the results provide a solid mathematical foundation for stochastic nonlocal diffuse-interface models and their local limits, including explicit rates under suitable regularity assumptions and mass-conserving properties.

Abstract

In this work, we deal with the stochastic counterpart of the nonlocal Cahn-Hilliard equation with regular potential in a smooth bounded two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of multiplicative type. For the resulting system, we are able to show the existence of probabilistically-weak (or martingale) solutions in two and three dimensions, that are unique and probabilistically-strong under suitable assumptions on the stochastic diffusion. Moreover, we investigate the nonlocal-to-local asymptotics toward solutions of the local stochastic Cahn-Hilliard equations, establishing, under regularity conditions, a precise rate of convergence as well.

The stochastic nonlocal Cahn-Hilliard equation with regular potential and multiplicative noise

TL;DR

The paper rigorously analyzes the stochastic nonlocal Cahn–Hilliard equation with a regular potential under multiplicative noise on bounded domains, proving existence of martingale solutions and, under Lipschitz diffusion, pathwise uniqueness and the existence of a probabilistically-strong solution. It develops a two-stage approximation (Yosida regularization plus Galerkin discretization) and employs stochastic compactness to pass to the limit, first in the Galerkin dimension and then in the regularization parameter, carefully identifying nonlinearities and stochastic integrals. The work also establishes nonlocal-to-local convergence, deriving rates under higher regularity and showing convergence without extra regularity via ε-uniform estimates and Gyöngy–Krylov methods. Altogether, the results provide a solid mathematical foundation for stochastic nonlocal diffuse-interface models and their local limits, including explicit rates under suitable regularity assumptions and mass-conserving properties.

Abstract

In this work, we deal with the stochastic counterpart of the nonlocal Cahn-Hilliard equation with regular potential in a smooth bounded two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of multiplicative type. For the resulting system, we are able to show the existence of probabilistically-weak (or martingale) solutions in two and three dimensions, that are unique and probabilistically-strong under suitable assumptions on the stochastic diffusion. Moreover, we investigate the nonlocal-to-local asymptotics toward solutions of the local stochastic Cahn-Hilliard equations, establishing, under regularity conditions, a precise rate of convergence as well.
Paper Structure (32 sections, 21 theorems, 281 equations)

This paper contains 32 sections, 21 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Functional setting and notation
  4. Stochastic setting.
  5. Analytical setting.
  6. Structural assumptions
  7. Main results
  8. Proof of Theorem \ref{['thm:existenceM']}
  9. The approximation scheme.
  10. Uniform estimates with respect to $n$
  11. First estimate.
  12. Second estimate.
  13. Third estimate.
  14. Fourth estimate.
  15. Passage to the limit as $n \to+\infty$
  16. ...and 17 more sections

Key Result

Theorem 2.4

Let Assumptions hyp:K-hyp:G hold. Let $p > 2$ and let $\varphi_0$ satisfy Then, there exists at least one martingale solution to the stochastic nonlocal Cahn--Hilliard equation eq:strongNCH.

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 25 more