Stability of the stationary profiles of the Allen-Cahn equation with not constant stiffness (original) (raw)

On an anisotropic Allen-Cahn system

Cubo (Temuco), 2015

Our aim in this paper is to prove the existence and uniqueness of solutions for an Allen-Cahn type system based on a modification of the Ginzburg-Landau free energy proposed in [11]. In particular, the free energy contains an additional term called Willmore regularization and takes into account anisotropy effects. RESUMEN Nuestro propósito en este trabajo es probar la existencia y unicidad de soluciones para un Sistema de tipo Allen-Cahn basados en una modificación de la energía libre Ginzburg-Landau propuesta en [11]. En particular, la energía libre contiene un término adicional llamado regularización de Willmore y considera efectos de anisotropía.

The non-isothermal Allen-Cahn equation with dynamic boundary conditions

We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter ψ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature θ. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.

An unconditionally gradient stable numerical method for solving the Allen–Cahn equation

Physica A: Statistical Mechanics and its Applications, 2009

We consider an unconditionally gradient stable scheme for solving the Allen-Cahn equation representing a model for anti-phase domain coarsening in a binary mixture. The continuous problem has a decreasing total energy. We show the same property for the corresponding discrete problem by using eigenvalues of the Hessian matrix of the energy functional. We also show the pointwise boundedness of the numerical solution for the Allen-Cahn equation. We describe various numerical experiments we performed to study properties of the Allen-Cahn equation.

A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions

Nonlinear Analysis, 2017

In this paper, we consider a class of coupled systems of PDEs, denoted by (ACE) ε for ε ≥ 0. For each ε ≥ 0, the system (ACE) ε consists of an Allen-Cahn type equation in a bounded spacial domain Ω, and another Allen-Cahn type equation on the smooth boundary Γ := ∂Ω, and besides, these coupled equations are transmitted via the dynamic boundary conditions. In particular, the equation in Ω is derived from the nonsmooth energy proposed by Visintin in his monography "Models of phase transitions": hence, the diffusion in Ω is provided by a quasilinear form with singularity. The objective of this paper is to build a mathematical method to obtain meaningful L 2-based solutions to our systems, and to see some robustness of (ACE) ε with respect to ε ≥ 0. On this basis, we will prove two Main Theorems 1 and 2, which will be concerned with the well-posedness of (ACE) ε for each ε ≥ 0, and the continuous dependence of solutions to (ACE) ε for the variations of ε ≥ 0, respectively.

Well-Posedness for One-Dimensional Anisotropic Cahn-Hilliard and Allen-Cahn Systems

2015

Our aim is to prove the existence and uniqueness of solutions for one-dimensional Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [8]. In particular, the free energy contains an additional term called Willmore regularization and takes into account strong anisotropy effects.

The Allen-Cahn equation with dynamic boundary conditions and mass constraints

Mathematical Methods in the Applied Sciences, 2014

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.