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Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

Helmut Abels, Yutaka Terasawa

TL;DR

The paper develops a rigorous framework for nonlocal-to-local convergence in singular, anisotropic nonlocal Cahn-Hilliard equations. By permitting nonradially symmetric, potentially nonintegrable kernels, it proves existence of weak solutions to the nonlocal problem for small ε and demonstrates convergence to a local anisotropic CH equation with μ = - div(A ∇c) + f'(c), where A is given by the kernel’s second moments. The authors establish key convergence of nonlocal energy and bilinear forms to their local counterparts, and apply the results to a diffuse-interface Navier–Stokes/Cahn–Hilliard system, showing convergence to the local, anisotropic NS/CCH model. The work broadens the scope of nonlocal diffuse-interface analysis to include anisotropy and singular kernels, with implications for two-phase flow modeling and related applications.

Abstract

We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels.

Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

TL;DR

The paper develops a rigorous framework for nonlocal-to-local convergence in singular, anisotropic nonlocal Cahn-Hilliard equations. By permitting nonradially symmetric, potentially nonintegrable kernels, it proves existence of weak solutions to the nonlocal problem for small ε and demonstrates convergence to a local anisotropic CH equation with μ = - div(A ∇c) + f'(c), where A is given by the kernel’s second moments. The authors establish key convergence of nonlocal energy and bilinear forms to their local counterparts, and apply the results to a diffuse-interface Navier–Stokes/Cahn–Hilliard system, showing convergence to the local, anisotropic NS/CCH model. The work broadens the scope of nonlocal diffuse-interface analysis to include anisotropy and singular kernels, with implications for two-phase flow modeling and related applications.

Abstract

We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels.

Paper Structure

This paper contains 5 sections, 5 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Preliminaries
  3. Existence of Solutions
  4. Nonlocal to Local Limit
  5. Application to a Navier-Stokes/Cahn-Hilliard System

Key Result

Lemma 2.3

For every $\varphi, \zeta\in H^1(\Omega)$ it holds that Moreover, the matrix $A$ is symmetric and positive definite. Furthermore, for every sequence $(\varphi_n)_{n \in \mathbb{N}}\subseteq L^2(\Omega)$, $\varepsilon_n>0$, $n\in\mathbb{N}$, tending to zero and $\varphi \in L^2(\Omega)$ it holds that

Theorems & Definitions (13)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Definition 5.1
  • Remark 5.2
  • ...and 3 more