We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing c...

We consider a diffusive interface surface tension model under compressible flow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field. These are two reduced models which show important properties of the full-scale surface tension model. We prove that both model problems are well-posed. We are especially interested in the beh...

We present an existence result for the Cahn{Hilliard equation with a concentration dependent mobility which allows the mobility to degenerate. Formal asymptotic results relate the Cahn{Hilliard equation with a degenerate mobility to motion by surface diiusion V = ? S. We state a local existence result for this geometric motion and show that circles are asymptotically stable.

The aim of this paper is to study the well-posedness and long time behavior, in terms of finite-dimensional attractors, of a perturbed Cahn–Hilliard equation. This equation differs from the usual Cahn–Hilliard by the presence of the term ε(−Δu+ f (u)). In particular, we prove the existence of a robust family of exponential attractors as ε goes to zero.

Abstract. We develop a new infinite dimensional gluing method for fractional elliptic equations. In Part I, as a model problem, we construct a solution of the fractional Allen–Cahn equation vanishing on a rotationally symmetric surface which resembles a catenoid and has sub-linear growth at infinity. In Part II, we construct counterexamples to De Giorgi Conjectures to fractional Allen-Cahn equa...

We propose a novel second order in time, decoupled and unconditionally stable numerical scheme for solving the Cahn-Hilliard-Darcy (CHD) system which models two-phase flow in porous medium or in a Hele-Shaw cell. The scheme is based on the ideas of second order convex-splitting for the Cahn-Hilliard equation and pressure-correction for the Darcy equation. We show that the scheme is uniquely sol...

A discontinuous Galerkin finite element method has been developed to treat the high-order spatial derivatives appearing in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear parabolic partial differential equation, originally proposed to model phase segregation of binary alloys. The developed discontinuous Galerkin approach avoids the need for mixed finite elemen...

The discrete Allen-Cahn equation is a spatially discrete analogue of the Allen-Cahn equation, a parabolic partial differential equation proposed as a simple model for phase separation in materials. In some sense, the solutions of the discrete equation display a richer variety of behaviors than do the corresponding solutions of the continuous equation. In particular, the number of stationary sol...

We consider Hamilton–Jacobi equations which characterize optimal controlled partial differential equations of the following types: the Allen–Cahn equation, the Cahn–Hilliard equation, a nonlinear Fokker–Planck equation, and aVlasov–Fokker–Planck equation. In each of the examples, the optimal control problem and its associated cost functional can be derived as limit from a microscopically define...

Spatially localized, time-periodic structures are common in pattern-forming systems, appearing in fluid mechanics, chemical reactions, and granular media. We examine the existence of oscillatory localized states in a PDE model with single frequency time dependent forcing, introduced in [22] as phenomenological model of the Faraday wave experiment. In this study, we reduce the PDE model to the f...