Résumé. Nous considérons l’équation de Kuramoto–Sivashinskymunie de conditions aux limites périodiques et d’une donnée initiale. Nous l’approchons en utilisant une méthode d’éléments finis de type Galerkin pour la discrétisation en espace, et un schéma de Runge–Kutta implicite pour la discrétisation en temps. Nous obtenons des estimations d’erreur optimales et discutons de la linéarisation de c...

We study stationary periodic solutions of the Kuramoto-Sivashinsky (KS) model for complex spatiotemporal dynamics in the presence of an additional linear destabilizing term. In particular, we show the phase space origins of the previously observed stationary “viscous shocks” and related solutions. These arise in a reversible four-dimensional dynamical system as perturbed heteroclinic connection...

We study the Kuramoto-Sivashinsky equation (KSE) in scalar form on two-dimensional torus with and without advection by an incompressible vector field. prove local existence of mild solutions for arbitrary data L2. then issue global existence. KSE presence data, provided advecting velocity field v satisfies certain conditions that ensure dissipation time associated hyperdiffusion-advection is su...

Recent work on the dynamics of a crystal surface [T. Frisch and A. Verga, Phys. Rev. Lett. 96, 166104 (2006)] has focused the attention on the conserved KuramotoSivashinsky (CKS) equation: ∂tu = −∂xx(u+ uxx + ux), which displays coarsening. For a quantitative and qualitative understanding of the dynamics, the analysis of steady states is particularly relevant. In this paper we provide a detaile...

We present a new topological method for the study of the dynamics of dissipative PDE’s. The method is based on the concept of the selfconsistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE’s subject to rigorously controlled small perturbation from the neglected modes. To this ODE’s we apply the Conley index to...

This paper deals with the null-controllability of a system mixed parabolic-elliptic pdes at any given time T>0. More precisely, we consider Kuramoto-Sivashinsky–Korteweg-de Vries equation coupled second order elliptic posed in interval (0,1). We first show that linearized is globally null-controllable by means localized interior control acting on either KS-KdV or equation. Using Carleman approa...