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.
Highlights
The aim of the present paper is to study the well-posedness and the asymptotic behaviour as ε 0 of a family of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions in the following form:
The main novelty of the present paper is to extend some rigorous nonlocal-tolocal convergence results for Cahn–Hilliard equations to the case of homogeneous Neumann boundary conditions
We define the family of convolution kernels as
Summary
The aim of the present paper is to study the well-posedness and the asymptotic behaviour as ε 0 of a family of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions in the following form:. Where (ρε)ε is a suitable family of mollifiers converging to a Dirac delta Building upon these variational convergences, in a previous contribution of ours [24] we rigorously derived some nonlocal-to-local asymptotics of solutions to Cahn–Hilliard equations in the setting of periodic boundary conditions and with no viscosity effects. The main consequence is that in the case of Neumann boundary conditions one loses any H 1-estimate on the nonlocal solutions It follows that the natural variational setting to frame the nonlocal problem (1.1)–(1.4) is not the usual one given by the triple (H 1( ), L2( ), H 1( )∗), but instead an abstract one (Vε, L2( ), Vε∗), depending on ε, where Vε represents, roughly speaking, the domain of the nonlocal energy contribution in (1.5).
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