Local asymptotics for nonlocal convective Cahn-Hilliard equations with W^{1,1} kernel and singular potential

TL;DR

This paper establishes the existence and analyzes the asymptotic behavior of solutions to nonlocal convective Cahn-Hilliard equations with W^{1,1} kernels and singular potentials, including various boundary conditions and extensions.

Contribution

It provides new results on existence and nonlocal-to-local asymptotics for a broad class of potentials and boundary conditions in nonlocal Cahn-Hilliard models.

Findings

Proved existence of solutions for the nonlocal convective Cahn-Hilliard equations.

Analyzed the nonlocal-to-local asymptotic behavior as the kernel becomes local.

Extended results to periodic boundary conditions and viscosity effects.

Abstract

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.