In one spatial dimension, the metastable dynamics and coarsening process of an n-layer pattern of internal layers is studied for the Cahn-Hilliard equation, the viscous Cahn-Hilliard equation, and the constrained Allen-Cahn equation. These models from the continuum theory of phase transitions provide a caricature of the physical process of the phase separation of a binary alloy. A homotopy para...

We build a family of entire solutions to the Allen-Cahn equation in RN+1 for N ≥ 3, whose level set approaches the higher dimensional catenoid in a compact region and has two logarithmic ends governed by the solutions to the Liouville equation.

We analyze a semidiscrete scheme for the Cahn-Hilliard equation in one space dimension, when the interface length parameter is equal to zero. We prove convergence of the scheme for a suitable class of initial data, and we identify the limit equation. We also characterize the long-time behavior of the limit solutions. keywords: Nonconvex functionals, forward-backward parabolic equations, finite ...

We study an Allen–Cahn equation perturbed by a multiplicative stochastic noise that is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive a uniform bound for the diffuse surface area, prove the tightness of solutions in the sharp interface limit, and show the convergence to phase-indicator functions.

We investigate a multiphase Cahn–Hilliard model for tumor growth with general source terms. The approach allows us to consider multiple cell types and chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared classical two-phase models, can be used describe stratified exhibiting several layers of tissue (e.g., proliferating, quiescent necrotic tissue) more precisely. O...

In this paper we study a model for phase segregation consisting in a sistem of a partial and an ordinary differential equation. By a careful definition of maximal solution to the latter equation, this system reduces to an Allen-Cahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the ω-limit set is given.

The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main part ...

We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nongenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem. Résumé. Nous étudions l’équat...

We present a nonlinear model predictive framework for closedloop control of two-phase flows governed by the Cahn-Hilliard NavierStokes system. We adapt the concept for instantaneous control from [6, 12, 16] to construct distributed closed-loop control strategies for twophase flows. It is well known that distributed instantaneous control is able to stabilize the Burger’s equation [16] and also t...