Time periodic solutions of Cahn-Hilliard systems with dynamic boundaryconditions

Abstract

The existence problem for Cahn-Hilliard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point theorem in the level of approximate ploblem. One of the key point is the assumption for maximal monotone graphs with respect to their domains. Thanks to this, we obtain the existence result of the weak solution by using the passage to the limit.

Weak Solutions of the Cahn--Hilliard System with Dynamic Boundary Conditions: A Gradient Flow Approach

“…In the case κ = 0 the problem is related to the moving contact line problem that is studied in [32]. Moreover, several dynamic boundary conditions have been proposed in the literature to replace the homogeneous Neumann conditions (see, e.g., [8,10,13,23,25,26,27,29,34]). Results for the Allen-Cahn equation (which is another phase-field equation, cf.…”

On the nonlocal Cahn–Hilliard equation with nonlocal dynamic boundary condition and boundary penalization

“…In the case κ = 0 this energy is related to the moving contact line problem (see, e.g., [70]). Recently, various dynamic boundary conditions corresponding to the energy E GL have been derived and analyzed in the literature, for instance [11,12,[16][17][18][19]35,37,38,44,56,[61][62][63][64]66,71,72]. In particular, Cahn-Hilliard systems with dynamic boundary conditions exhibiting also a Cahn-Hilliard type structure have become very popular in recent times.…”

Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn-Hilliard type with singular potential

“…Concerning the problem with general (singular) potentials and non-permeable wall (κ = 0), existence and uniqueness of global weak solutions and their long-time behavior were studied in [6] (σ ≥ 0, χ ≥ 0) and [25] (σ = χ = 1), see also [9] in which the double obstacle potential was handled and recent works [11,12,23] for the system with additional convection and viscous terms. Last but not least, we refer to [7,17] for numerical studies, to [16] for the associated optimal boundary control problem, and to [41] for the existence of time periodic solutions.…”