On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions

Abstract

The Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions is considered and well-posedness results are proved.

The Cahn-Hilliard Equation with Logarithmic Potentials

“…Our aim in this article is to discuss the Cahn-Hilliard system with a logarithmic (or a more general singular) nonlinear term f (as noted above, in the case of dynamic boundary conditions, the surface nonlinear term g is expected to be regular in relevant situations, although one could also consider a singular surface nonlinearity, see [118]). More details (and, in particular, the proofs of the results stated below) can be found in [169] and [173].…”

“…The first proof of existence of weak energy solutions to the Cahn-Hilliard equation with singular (and, in particular, logarithmic) potentials and dynamic boundary conditions is given in [118], assuming that the (regular) surface nonlinearity g has the right sign at the singular points of the bulk nonlinearity f , namely,…”

Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions

“…The Cahn-Hilliard system, endowed with these boundary conditions, has been studied in [6,18,23,24,26,27] (see also [4-6, 12-15, 17] for similar boundary conditions for the Caginalp phase-field system). Now, while the problem is well-understood for regular nonlinear terms f 0 and f Γ , in the sense that we have rather complete and satisfactory results concerning the well-posedness, the regularity of the solutions and the asymptotic behavior of the system (namely, the existence of finite-dimensional attractors and the convergence of trajectories to steady states), the situation is less clear for an irregular nonlinear bulk term f 0 , and, in particular, for the above logarithmic function.…”

Existence and Uniqueness of Solution for Cahn-Hilliard Hyperbolic Phase-Field System with Dirichlet Boundary Condition and Regular Potentials

“…These equations model phase transition processes such as melting-solidification processes and have been studied, see [2]- [6], for a similar phase-field model with a nonlinear term. These Cahn-Hilliard phase-fiel system are known as the conserved phase-field system (see [7]- [9]) based on type III heat conduction and with two temperatures (see [10]).…”