Maria Gokieli

Global attractor for the Cahn--Hilliard/Allen--Cahn system

We study the coupled Cahn-Hilliard/Allen-Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence-uniqueness result and to construct the global attractor for the related dynamical system.

Discrete Approximation of the Cahn-Hilliard/Allen-Cahn System with Logarithmic Entropy

We propose a numerical scheme for the Cahn-Hilliard/Allen-Cahn system with logarithmic nonlinearity, based on the finite element method. We show its well-posedeness and convergence of the numerical solution to the weak solution of the original system. We point out some properties of a regularized numerical solution.

The Neumann problem in an irregular domain

We ask the question of patterns’ stability for the reaction-diffusion equation with Neumann boundary conditions in an irregular domain in R , N ≥ 2, the model example being two convex regions connected by a small ’hole’ in their boundaries. By patterns we mean solutions having an interface, i.e. a transition layer between two constants. It is well known that in 1D domains and in many 2D domains ’patterns’ are unstable for this equation. We show that, unlike the 1D case, but as in 2D dumbbell domains, stable patterns exist. In a more general way, we prove invariance of stability properties for steady states when a sequence of domains Ωn converges to our limit domain Ω in the sense of Mosco. We illustrate the theoretical results by numerical simulations of evolving and persisting interfaces. ∗To whom correspondence should be addressed

Variational inequalities of Navier–Stokes type with time dependent constraints

Abstract We consider a class of parabolic variational inequalities with time dependent obstacle of the form | u ( x , t ) | ≤ p ( x , t ) , where u is the velocity field of a fluid governed by the Navier–Stokes variational inequality. The obstacle function p = p ( x , t ) , imposed on u, consists of three parts, which are respectively: the degenerate part p ( x , t ) = 0 , the finitely positive part 0 p ( x , t ) ∞ and the singular part p ( x , t ) = ∞ . In this paper, we shall propose a sequence of approximate obstacle problems with everywhere finitely positive obstacles, and prove an existence result for the original problem by discussing convergence of the approximate problems. The crucial step is to handle the nonlinear convection term. In this paper we propose a new approach to it.

Simulating Phase Transition Dynamics on Non-trivial Domains

Our goal is to investigate the influence of the geometry and topology of the domain \(\varOmega \) on the solutions of the phase transition and other diffusion-driven phenomena in \(\varOmega \), modeled e.g. by the Allen–Cahn, Cahn–Hilliard, reaction–diffusion equations. We present FEM numerical schemes for the Allen–Cahn and Cahn–Hilliard equation based on the Eyre’s algorithm and present some numerical results on split and dumbbell domains.