We present numerical simulations which support the formal asymptotic analysis relating a multi order parameter Allen{Cahn system to a multi phase interface problem with curvature dependent evolution of the interfaces and angle conditions at triple junctions. Within the gradient energy of the Allen{Cahn system, the normal to an interface between phases i and j is modeled by the irreducible repre...

Two-Phase Segmentation for Intensity Inhomogeneous Images by the Allen--Cahn Local Binary Fitting Model

<p style='text-indent:20px;'>We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by Caputo derivative. Several novel criterions are examined showing positive-definiteness associated kernel functions. deduce strict energy-dissipation number non-local energy fu...

Weakly nonlocal thermodynamic theories are critically revisited. The irreversible thermodynamic theory of nonlocal phenomena is given, based on a modified form of the entropy current. Several classical equations are derived , including Guyer-Krumhansl, Ginzburg-Landau and Cahn-Hilliard type equations.

We derive a new, effective macroscopic Cahn–Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn–Hilliard equation in such domains. The ...

We study the asymptotic behaviour of radially symmetric solutions of the nonlocal equation In a bounded spherically symmetric domain ft C R"f where A,(<) m 1 fftW(<p) «fcr, with a Neumann boundary condition. The analysis is based on "energy methods combined with some a-priori estimates, the latter being used to approximate the sohtion by the first two terms of an asymptotic expansionWe only nee...

The goal of this paper is to study the slow motion of solutions of the nonlocal Allen–Cahn equation in a bounded domain Ω ⊂ R, for n > 1. The initial data is assumed to be close to a configuration whose interface separating the states minimizes the surface area (or perimeter); both local and global perimeter minimizers are taken into account. The evolution of interfaces on a time scale ε−1 is d...

We consider a diffuse interface model for tumour growth consisting of a Cahn– Hilliard equation with source terms coupled to a reaction-diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transpo...

This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. The free energy density associated to the Cahn-Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time insta...

We propose and analyze two novel decoupled numerical schemes for solving the Cahn-HilliardStokes-Darcy (CHSD) model for two-phase flows in karstic geometry. In the first numerical scheme, we explore a fractional step method (operator splitting) to decouple the phase-field (Cahn-Hilliard equation) from the velocity field (Stokes-Darcy fluid equations). To further decouple the Stokes-Darcy system...