Asymptotic limits and optimal control for the Cahn-Hilliard system with convection and dynamic boundary conditions

TL;DR

This paper investigates the long-term behavior and optimal control of a nonviscous Cahn-Hilliard system with convection and dynamic boundary conditions, extending previous viscous case results by analyzing the zero-viscosity limit.

Contribution

It provides the first analysis of the nonviscous Cahn-Hilliard system with dynamic boundary conditions, including asymptotic behavior and optimal control results derived from viscous case estimates.

Findings

Characterization of omega-limit sets as stationary solutions

Existence of optimal controls for the fluid velocity

Uniform estimates allowing zero-viscosity limit transition

Abstract

In this paper, we study initial-boundary value problems for the Cahn--Hilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous Cahn--Hilliard systems, the `pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the $\omega$-limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and first-order necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case.