Stability of the travelling wave in a 2D weakly nonlinear Stefan problem

Pages: 109 - 134, Volume 2, Issue 1, March 2009

doi:10.3934/krm.2009.2.109       Abstract        Full Text (328.4K)       Related Articles

Claude-Michel Brauner - Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence cedex, France (email)
Josephus Hulshof - Faculty of Sciences – Mathematics and Computer Science division, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081HV Amsterdam, Netherlands (email)
Luca Lorenzi - Dipartimento di Matematica, Universitá degli Studi di Parma, Viale G. Usberti 85/A, 43100 Parma, Italy (email)

Abstract: 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.

Keywords:  Stefan problem, stability, front dynamics, Kuramoto-Sivashinsky equation, pseudo-differential operators, sectorial operators
Mathematics Subject Classification:  Primary: 35K55; Secondary: 35B35, 80A22.

Received: September 2008;      Revised: November 2008;      Published: January 2009.