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March  2009, 2(1): 109-134. doi: 10.3934/krm.2009.2.109

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

Claude-Michel Brauner 1, , Josephus Hulshof 2, and  Luca Lorenzi 3,

1. 

Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence cedex
2. 

Faculty of Sciences – Mathematics and Computer Science division, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081HV Amsterdam
3. 

Dipartimento di Matematica, Universitá degli Studi di Parma, Viale G. Usberti 85/A, 43100 Parma

Received  September 2008 Revised  November 2008 Published  January 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.
Keywords: Stefan problem, stability, Kuramoto-Sivashinsky equation, sectorial operators, front dynamics, pseudo-differential operators.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B35, 80A2.
Citation: Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109
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