Stability of the travelling wave in a 2D weakly nonlinear Stefan problem
Claude-Michel Brauner - Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 33405 Talence cedex, France (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
Received: September 2008; Revised: November 2008; Published: January 2009.
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