Georgia Karali

Front Motion in the One-Dimensional Stochastic Cahn--Hilliard Equation

In this paper, we consider the one-dimensional Cahn--Hilliard equation perturbed by additive noise and study the dynamics of interfaces for the stochastic model. The noise is smooth in space and defined as a Fourier series with independent Brownian motions in time. Motivated by the work of Bates and Xun on slow manifolds for the integrated Cahn--Hilliard equation, our analysis reveals the significant difficulties and differences in comparison to the deterministic problem. New higher order terms that we estimate appear due to Ito calculus and stochastic integration and dominate the exponentially slow deterministic dynamics. Using a local coordinate system and defining the admissible interface positions as a multidimensional diffusion process, we derive a first order linear system of stochastic ordinary differential equations approximating the equations of front motion. Furthermore, we prove stochastic stability of the approximate slow manifold of solutions over a very long time scale and evaluate the noise...

Pulsating Wave for Mean Curvature Flow in Inhomogeneous Medium

We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

Motion of bubbles towards the boundary for the Cahn–Hilliard equation

In this work we describe some aspects of the dynamics of the Cahn–Hilliard equation. In particular, we consider the dynamics of ‘bubble’ solutions that is spherical interfaces which move superslowly towards the boundary without changing their shape. We show for the Cahn–Hilliard that the bubble drifts towards the closest point on the boundary provided it is sufficiently small. This is contrasted with the related mass conserving Allen–Cahn equation where size is not an issue.

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

We consider the following singularly perturbed elliptic problem: where Ω is a bounded domain in ℝ2 with smooth boundary, ϵ > 0 is a small parameter, n denotes the outward normal of ∂Ω, and a, b are smooth functions that do not depend on ϵ. We assume that the zero set of a − b is a simple closed curve Γ, contained in Ω, and ∇(a − b) ≠ 0 on Γ. We will construct solutions u ϵ that converge in the Hölder sense to max {a, b} in Ω, and their Morse index tends to infinity, as ϵ → 0, provided that ϵ stays away from certain critical numbers. Even in the case of stable solutions, whose existence is well established for all small ϵ > 0, our estimates improve previous results.

Singular Limit of a Spatially Inhomogeneous Lotka–Volterra Competition–Diffusion System

We discuss the generation and the motion of internal layers for a Lotka-Volterra competition-diffusion system with spatially inhomogeneous coefficients. We assume that the corresponding ODE system has two stable equilibria (ū, 0) and (0, ) with equal strength of attraction in the sense to be specified later. The equation involves a small parameter ϵ, which reflects the fact that the diffusion is very small compared with the reaction terms. When the parameter ϵ is very small, the solution develops a clear transition layer between the region where the u species is dominant and the one where the v species is dominant. As ϵ tends to zero, the transition layer becomes a sharp interface, whose motion is subject to a certain law of motion, which is called the “interface equation”. A formal asymptotic analysis suggests that the interface equation is the motion by mean curvature coupled with a drift term. We will establish a rigorous mathematical theory both for the formation of internal layers at the initial stage and for the motion of those layers in the later stage. More precisely, we will show that, given virtually arbitrary smooth initial data, the solution develops an internal layer within the time scale of O(ϵ2logϵ) and that the width of the layer is roughly of O(ϵ). We will then prove that the motion of the layer converges to the formal interface equation as ϵ → 0. Our results also give an optimal convergence rate, which has not been known even for spatially homogeneous problems.

Continuum limits of particles interacting via diffusion

We consider a two-phase system mainly in three dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasistatic Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean-field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer andOtto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking into account the geometry of the spatial distribution.

Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain

First published in Mathematics of Computation online 2014 (84 (2015), 1571-1598), published by the American Mathematical Society

A Hilbert expansion method for the rigorous sharp interface limit of the generalized Cahn-Hilliard equation

We consider Cahn-Hilliard equations with external forcing terms. Energy decreasing and mass conservation might not hold. We show that level surfaces of the solutions of such generalized Cahn-Hilliard equations tend to the solutions of a moving boundary problem under the assumption that classical solutions of the latter exist. Our strategy is to construct approximate solutions of the generalized Cahn-Hilliard equation by the Hilbert expansion method used in kinetic theory and proposed for the standard Cahn-Hilliard equation, by Carlen, Carvalho and Orlandi, \cite {CCO}. The constructed approximate solutions allow to derive rigorously the sharp interface limit of the generalized Cahn-Hilliard equations. We then estimate the difference between the true solutions and the approximate solutions by spectral analysis, as in \cite {A-B-C}

MOTION OF A DROPLET FOR THE STOCHASTIC MASS-CONSERVING ALLEN-CAHN EQUATION ∗

We study the stochastic mass-conserving Allen--Cahn equation posed on a smoothly bounded domain of

The sharp interface limit for the stochastic Cahn-Hilliard Equation

\mathbb{R}^2