We consider the spectrum associated with the linear operator obtained when a Cahn-Hilliard system on Rn is linearized about a planar transition front solution. In the case of single Cahn-Hilliard equations on Rn, it’s known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate |ξ|3, where ξ is the Fourier transform variable corresponding...

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short tim...

The inpainting of damaged images has a wide range of applications and many different mathematical methods have been proposed to solve this problem. Inpainting with the help of Cahn-Hilliard models has been particularly successful, and it turns out that Cahn-Hilliard inpainting with the double obstacle potential can lead to better results compared to inpainting with a smooth double well potentia...

At high temperature and pressure, solid diffusion chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill causes separation, that is, spinodal decomposition. Thus, describe the coarsening kinetics of exsolution microstructure, we derive a thermodynamically consistent continuum theory for multicomponent Cahn--Hilliard equations while accounting mu...

We consider a generalisation of the Cahn–Hilliard equation that incorporates an elastic energy density which, being quasiconvex, incorporates microstructure formation on smaller length scales. We prove global existence of weak solutions in certain microstructural regimes in (one and) two dimensions and present sufficient conditions for uniqueness. Preliminary numerical computations to illustrat...

In this paper, we established traveling wave solutions of the nonlinear evolution equation. The sine-cosine method was used to construct travelling wave solutions of the Fitzhugh-Nagumo equation and Cahn-Allen equation. Graphical interpretation shows that obtained results include periodic and Soliton wave solutions. It is also shown that for α =−1, solutions of Fitzhugh-Nagumo equation coincide...

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 consider the Allen-Cahn equation ε∆u + u− u = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where Ω is a smooth and bounded domain in R such that the mean curvature is positive at each boundary point. We show that there exists a sequence εj → 0 such that the Allen-Cahn equation has a solution uεj with an interface which approaches the boundary as j → +∞.

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper. Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed, and are shown to satisfy discrete energy laws which are analogou...