[Paper Review] From nonlocal to local Cahn-Hilliard equation

Stefano Melchionna, Helene Ranetbauer+2 more|arXiv (Cornell University)|Mar 26, 2018

Solidification and crystal growth phenomena31 references17 citations

TL;DR

This paper establishes the convergence of weak solutions to the nonlocal Cahn-Hilliard equation to those of the local Cahn-Hilliard equation as the nonlocal kernel approximates a Dirac delta in a periodic domain. The proof relies on the H⁻¹-gradient flow structure, uniform H¹ estimates, and a Poincaré-type inequality by Ponce to ensure compactness and strong convergence in the limit.

ABSTRACT

In this paper we prove the convergence of a nonlocal version of the Cahn-Hilliard equation to its local counterpart as the nonlocal convolution kernel is scaled using suitable approximations of a Dirac delta in a periodic boundary conditions setting. This convergence result strongly relies on the dynamics of the problem. More precisely, the $H^{-1}$-gradient flow structure of the equation allows to deduce uniform $H^1$ estimates for solutions of the nonlocal Cahn-Hilliard equation and, together with a Poincaré type inequality by Ponce, provides the compactness argument that allows to prove the convergence result.

Motivation & Objective

• To rigorously justify the passage from nonlocal to local Cahn-Hilliard models in the context of phase separation and spinodal decomposition.
• To analyze the asymptotic behavior of weak solutions as the nonlocal interaction kernel converges to a Dirac delta.
• To establish uniform regularity and compactness in the nonlocal setting to ensure convergence to the local solution.
• To address the challenge of boundary conditions in the limit, particularly in the absence of direct control on u at the boundary.
• To extend the understanding of gradient flow structures in nonlocal and local phase-field models under periodic boundary conditions.

Proposed method

• Utilizes the H⁻¹-gradient flow structure of the nonlocal Cahn-Hilliard equation to derive uniform H¹ estimates for solutions.
• Applies a Poincaré-type inequality by Ponce to control the H¹ seminorm and ensure compactness in the limit.
• Employs a family of convolution kernels $ K_\varepsilon(x,y) = \varepsilon^{-d}|x-y|^{-2}\rho\left(\frac{|x-y|}{\varepsilon}\right) $ that approximate the Dirac delta as $ \varepsilon \to 0 $.
• Uses variational formulations and subdifferential calculus to pass to the limit in the energy and flux terms.
• Applies the dominated convergence theorem and strong convergence in $ L^2(0,T;L^2) $ to show convergence of the nonlocal energy to the local energy.
• Establishes that the limit satisfies the local Cahn-Hilliard equation in a weak, integrated-in-time variational form.

Experimental results

Research questions

• RQ1 Can weak solutions of the nonlocal Cahn-Hilliard equation converge to weak solutions of the local Cahn-Hilliard equation as the nonlocal kernel approaches a Dirac delta?
• RQ2 What structural properties of the nonlocal equation (e.g., gradient flow, energy bounds) enable such convergence?
• RQ3 How does the Poincaré-type inequality by Ponce contribute to the compactness and regularity of the limit solution?
• RQ4 What are the implications for boundary conditions in the limit, particularly when only the chemical potential is constrained?
• RQ5 Can the convergence result be extended to other boundary conditions, such as Neumann or Dirichlet?

Key findings

• Weak solutions of the nonlocal Cahn-Hilliard equation converge strongly in $ L^2(0,T;H^1(Ω)) $ to a weak solution of the local Cahn-Hilliard equation as $ \varepsilon \to 0 $.
• The convergence is established via uniform $ H^1 $ estimates derived from the $ H^{-1} $-gradient flow structure and a Poincaré-type inequality.
• The limit solution $ u $ belongs to $ L^2(0,T;H^2(\Omega)) $, ensuring sufficient regularity for the weak formulation to be well-defined.
• The nonlocal energy $ \tilde{E}_\varepsilon(u_\varepsilon) $ converges to the local energy $ \tilde{E}(u) $ in $ L^1(0,T) $, which is essential for the limit passage.
• The limit satisfies the local Cahn-Hilliard equation in the weak, integrated-in-time variational form: $ \int_0^T \int_\Omega (\partial_t u)\varphi \,dx\,dt - \int_0^T \int_\Omega \nabla u \cdot \nabla \Delta\varphi \,dx\,dt - \int_0^T \int_\Omega F'(u) \Delta\varphi \,dx\,dt = 0 $ for all smooth test functions $ \varphi $.
• The result holds under periodic boundary conditions in dimension $ d=3 $, which are physically relevant for lattice-based derivations of the nonlocal model.