This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling motion an incompressible two-phase fluid mixture in presence capillarity effects a bounded smooth domain $\Omega\subset \mathbb{R}^d$, $d=2,3$. We focus on dissipative mixing originating from mass-conserving Allen-Cahn dynamics with physically relevant Flory-Huggins potential. More precisely, we study N...

We present in this note a unified approach on how to design simple, efficient and energy stable time discretization schemes for the Allen-Cahn or Cahn-Hilliard Navier-Stokes system which models twophase incompressible flows with matching or non-matching density. Special emphasis is placed on designing schemes which only require solving linear systems at each time step while satisfy discrete ene...

Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...

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 extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example of this type of equation which we analyze in some detail is the Cahn-Hilliard equation. We analyze the marginally stable solutions of this equation in some de...

Abstract: The paper concerns the existence of weak solutions to the Cahn-Hilliard-Gurtin system coupled with nonstationary elasticity. The system describes phase separation process in elastically stressed material. It generalizes the Cahn-Hilliard equation by admitting a more general structure and by coupling diffusive and elastic effects. The system is studied with the help of a singularly per...

Correction for 'Frustrated phases under three-dimensional confinement simulated by a set of coupled Cahn-Hilliard equations' by Edgar Avalos et al., Soft Matter, 2016, 12, 5905-5914.

The gradient flow approach to the Cahn-Hilliard and phase field models is developed, and some basic mathematical properties of the models, especially phase separation phenomena, are reviewed.

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...