Dimitra Antonopoulou

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...

Galerkin Methods for Parabolic and Schrödinger Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics

In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schrodinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the half-line

L^2

Conservative initial‐boundary value problems for the Wide‐Angle PE in waveguides with variable bottoms

and

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

H^1

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

are proved for the associated semidiscrete and fully discrete Crank-Nicolson-Galerkin approximations. The problem involving the Schrodinger equation is motivated by considering the standard “parabolic” (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long-range sound propagation in the sea, in the specific case of a domain with a rigid bottom of variable topography. This model is contrasted with alternative ones that avoid the dynamical bottom boundary condition and are shown to yield qualitatively better approximations. In the (real) parabolic case, numerical approximations are considered for dynamical boundary conditions of reactive and dissipative type.

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

This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0951-7715/28/9/3073

The sharp interface limit for the stochastic Cahn-Hilliard Equation

We consider the third‐order, Claerbout‐type Wide‐Angle Parabolic Equation (PE) in the context of Underwater Acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There are strong indications, that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition on B, may not be well‐posed, for example when B is downsloping. In previous work we proposed an additional bottom boundary condition that, together with the zero field condition on B, yields a well‐posed problem. In the present paper we continue our investigation of additional bottom boundary conditions that yield well‐posed, physically correct problems. Motivated by the fact that the solution of the wide‐angle PE in a domain with horizontal layers conserves its L2 norm in the absence of attenuation, we seek additional boundary conditions on a variable‐topography bottom, that yield L2‐ conservative solutions of the problem. We identify a family ...

Addendum to ‘On the Dirichlet to Neumann problem for the 1-dimensional cubic NLS equation on the half-line’

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

A NONLINEAR PARTIAL DIFFERENTIAL EQUATION FOR THE VOLUME PRESERVING MEAN CURVATURE FLOW

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}