Asymptotic Analysis

An integral representation result is obtained for the relaxation of a class of energy functionals depending on two vector fields with different behaviors which appear in the context of thermochemical equilibria and are related to image decomposition models and directors theory in nonlinear elasticity.

We show the H 1 scattering for a one dimensional nonlinear Schrödinger equation with a non-negative, repulsive potential V such that V , x V ∈ W 1 , 1 , and a mass-supercritical non-linearity. We follow the approach of concentration-compacity/rigidity first introduced by Kenig and Merle.

In this paper we consider a plate equation with internal feedback and viscoelastic damping localized on a part of the boundary. Without imposing restrictive assumptions on the time-dependent frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

We address the existence of solutions for the free-surface Euler equation with surface tension in a bounded domain. Considering the problem in Lagrangian variables we provide a priori estimates leading to existence of local solutions with the initial velocity in H 3.5 for which the trace on the free boundary belongs to H 3.5 .

In this paper, we present a simplified electrical model for tissue culture. We derive a mathematical structure for overall electrical properties of the culture and study their dependence on the frequency of the current. We introduce a method for recovering the microscopic properties of the cell culture from the spectral measurements of the effective conductivity. Numerical examples are provided to illustrate the performance of our approach.

This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path ε and the heterogeneity length scale η vanish. No periodicity assumption is made on the scattering coefficient of the background material. There is an assumption made on the heterogeneity length scale η that it scales as ε β for β ∈ ( 0 , ∞ ) . In one space dimension, we prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation for any β ⩽ 2 in the ε → 0 limit. In any arbitrary phase space dimension, under a smallness assumption of a certain quotient involving the scattering coefficient in the H − 1 2 norm, we again prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation in the ε → 0 limit.