Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms

Abstract

We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

References in corpus (12)

• Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits
• A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
• An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis
• A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis
• On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential
• Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics
• Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation
• From nonlocal to local Cahn-Hilliard equation
• On a nonlocal Cahn-Hilliard equation with a reaction term
• Existence and uniqueness of solutions to singular Cahn-Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions
• From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
• A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity

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• A Unified Theory of Free Energy Functionals and Applications to Diffusion
• The 3D strict separation property for the nonlocal Cahn-Hilliard equation with singular potential
• On the inviscid limit connecting Brinkman's and Darcy's models of tissue growth with nonlinear pressure