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

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.

💡 Research Summary

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This paper investigates the stochastic nonlocal Cahn–Hilliard equation with a regular (smooth) double‑well potential and multiplicative cylindrical noise in a smooth bounded domain of dimension two or three. The authors consider homogeneous Neumann boundary conditions and random initial data. The main contributions are threefold.

First, under a set of structural assumptions on the interaction kernel (K), the potential (F) and the diffusion operator (G) (symmetry and integrability of (K); (F\in C^{2}) with controlled growth and a coercivity condition (F’’(s)+a(x)\ge C_{0}>0); Lipschitz continuity, boundedness and linear growth of (G) in appropriate Hilbert spaces), the paper establishes the existence of martingale (probabilistically weak) solutions. The proof relies on a Yosida regularization of the nonlinear terms, a Galerkin approximation, and uniform energy estimates obtained via Itô’s formula. These estimates give bounds in (L^{p}(\Omega;L^{\infty}(0,T;H))) and (L^{2}(\Omega;L^{2}(0,T;V))). Using stochastic compactness (tightness, Skorokhod representation) the authors pass to the limit and obtain a martingale solution that conserves the total mass and satisfies an energy inequality.

Second, the paper proves pathwise uniqueness and consequently the existence of probabilistically strong solutions. Uniqueness is achieved by considering two solutions, forming their difference, and applying Itô’s formula to the squared (H)-norm of the difference. The Lipschitz property of (G) and the monotonicity of the deterministic part (thanks to the coercivity condition) lead to a Grönwall‑type estimate that forces the difference to vanish almost surely. By the Yamada–Watanabe theorem, the weak solution is upgraded to a strong one.

Third, the authors study the nonlocal‑to‑local limit. They introduce a family of rescaled kernels (K_{\varepsilon}(x)=\varepsilon^{-d}K(x/\varepsilon)) and consider the corresponding solutions (\varphi_{\varepsilon}). Using Fourier analysis of the convolution operator, they show that ((K_{\varepsilon}1)\varphi-K_{\varepsilon}\varphi) converges to (-\Delta\varphi) as (\varepsilon\to0). By comparing the energy identities for the nonlocal and local problems and exploiting the uniform bounds already obtained, they derive a quantitative convergence rate: \