On the longtime behavior of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions

TL;DR

This paper analyzes the long-term behavior of a viscous convective Cahn-Hilliard system with dynamic boundary conditions, demonstrating that solutions tend to stationary states under certain conditions.

Contribution

It extends previous work by considering dynamic boundary conditions and nondifferentiable free energies, providing new insights into the asymptotic behavior of the system.

Findings

Solutions' omega-limit sets are nonempty and consist of stationary solutions.

Under positive viscosity and differentiable free energy, solutions stabilize over time.

The chemical potential component approaches a constant in the long run.

Abstract

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were are assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energies functionals that may be nondifferentiable and have derivatives only in the sense of (possibly set-valued) subdifferentials. For the resulting initial-boundary value system of Cahn-Hilliard type, general well-posedness results have been established in a recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the $\omega$-limit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the $\omega$-limit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant.