765 articles – 1378 Notices  [english version]
 HAL : hal-00386986, version 1
 Kinetic and related models 2, 1 (2009) 109-134
 Stability of the travelling wave in a 2D weakly nonlinear Stefan problem
 (03/2009)
 We investigate the stability of the travelling wave (TW) solution in a 2D Stefan problem, a simplified version of a solid-liquid interface model. It is intended as a paradigm problem to present our method based on: (i) definition of a suitable linear one dimensional operator, (ii) projection with respect to the $x$ coordinate only; (iii) Lyapunov-Schmidt method. The main issue is that we are able to derive a parabolic equation for the corrugated front $\varphi$ near the TW as a solvability condition. This equation involves two linear pseudo-differential operators, one acting on $\varphi$, the other on $(\varphi_y)^2$ and clearly appears as a generalization of the Kuramoto-Sivashinsky equation related to turbulence phenomena in chemistry and combustion. A large part of the paper is devoted to study the properties of these operators in the context of functional spaces in the $y$ and $x,y$ coordinates with periodic boundary conditions. Technical results are deferred to the appendices.
 1 : Institut de Mathématiques de Bordeaux (IMB) CNRS : UMR5251 – Université Sciences et Technologies - Bordeaux I – Université Victor Segalen - Bordeaux II 2 : Department of Computer Science [Amsterdam] Vrije Universiteit Amsterdam
 Domaine : Mathématiques/Equations aux dérivées partielles
 Mots Clés : Stefan problem – stability – front dynamics – Kuramoto-Sivashinsky equation – pseudo-differential operators – sectorial operators
 hal-00386986, version 1 http://hal.archives-ouvertes.fr/hal-00386986 oai:hal.archives-ouvertes.fr:hal-00386986 Contributeur : Claude-Michel Brauner <> Soumis le : Vendredi 22 Mai 2009, 12:21:53 Dernière modification le : Vendredi 22 Mai 2009, 12:21:53