# On a Cahn–Hilliard system with convection and dynamic boundary conditions

@article{Colli2017OnAC, title={On a Cahn–Hilliard system with convection and dynamic boundary conditions}, author={Pierluigi Colli and Gianni Gilardi and J{\"u}rgen Sprekels}, journal={Annali di Matematica Pura ed Applicata (1923 -)}, year={2017}, volume={197}, pages={1445-1475} }

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn–Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn–Hilliard cases are investigated, and a number of results are proven about existence of solutions…

## 24 Citations

Global Existence for a Nonstandard Viscous Cahn-Hilliard System with Dynamic Boundary Condition

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- 2017

A model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli is studied, and a dynamic boundary condition involving the Laplace--Beltrami operator for the order parameter is considered.

Optimal Velocity Control of a Viscous Cahn-Hilliard System with Convection and Dynamic Boundary Conditions

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- 2018

This paper shows existence for the optimal control problem under investigation, proves the Fr\'echet differentiability of the associated control-to-state mapping in suitable Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system.

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

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

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AbstractIn 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…

The Cahn-Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity

- MathematicsSIAM J. Math. Anal.
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An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn–Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to…

Vanishing diffusion in a dynamic boundary condition for the Cahn–Hilliard equation

- MathematicsNonlinear Differential Equations and Applications NoDEA
- 2020

The initial boundary value problem for a Cahn-Hilliard system subject to a dynamic boundary condition of Allen-Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary…

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- 2021

We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The…

Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

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- 2021

A Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for…

Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A 'deep quench' approach

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In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes…

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

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- 2019

The Cahn–Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in…

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