Allen-Cahn Model Research Articles

Diffuse interface surface tension models in an expanding flow

We consider a di usive interface surface tension model under compressible ow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity eld. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more e ective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle.

The effective energy in the Allen–Cahn model with deformation

The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen–Cahn type with deformation, with a parameter \varepsilon measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for \varepsilon\to0 of the equilibrium energy is determined. The interface energy is defined as the excess energy over the corresponding two phase state with a sharp interface without the interface energy. It is shown that to within the term of order {\rm o}(\varepsilon) the interface energy is equal to \sigma\varepsilon where the coefficient \sigma is given by a new formula that involves the mechanical contribution to the total energy. Also the corresponding equilibrium states are determined and shown to converge to a sharp interface state for \varepsilon\to0.

Single and Multitransition Solutions for a Family of Semilinear Elliptic PDE’s

These lectures survey a class of partial differential equations whose study was initiated by Moser. This class includes various Allen-Cahn models of phase transitions. In line with this connection, the existence of a large variety of solutions which undergo spatial transitions is established. The main tools are variational arguments, especially minimization methods.

Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen–Cahn System

We consider the singular limit of a perturbed Allen–Cahn model on a bounded two-dimensional domain: $\left\{\begin{array}{@{}ll@{}} u_t = \varepsilon^2 \Delta u - 2 (u - \varepsilon a) (u^2 - 1), & x \in \Omega \subset \mathbb{R}^2 \ \partial_n u = 0, & x \in \partial \Omega \end{array} \right.$ where $\varepsilon$ is a small parameter and a is an $O(1)$ quantity. We study equilibrium solutions that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the time-dependent PDE as well as of the associated two-dimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.

Infinite transition solutions for a class of Allen–Cahn model equations

This paper constructs a family of solutions that undergoes an infinite number of spatial transitions for an Allen–Cahn model equation.

On a class of infinite transition solutions for an Allen-Cahn model equation

Using minimization arguments and a limit process, we construct a family of solutions which undergo an infinite number of transitions for an Allen-Cahn model equation.

Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids

The goal of this paper is to derive again the generalized Cahn-Hilliard andAllen-Cahn models in deformable continua introduced previously by E. Friedand M. E. Gurtin on the basis of a microforce balance. We use a~differentapproach based on the second law in the form of the entropy principleaccording to I. Müller and I. S. Liu which leads to the evaluation of theentropy inequality with multipliers. Both approaches provide the same systems of field equations. In particular, ourdifferential equation for the multiplier associated with the balance law forthe order parameter turns out to be identical with the Fried-Gurtinmicroforce balance.

Mixed states for an Allen-Cahn type equation, II

This paper continues the recent study of an Allen-Cahn model PDE [1] by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions, enabling us to obtain multibump solutions in a much more general setting. As in [1], novel minimization arguments play a key role in finding solutions.

Conformal curvature flows: From phase transitions to active vision

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.