Rigorous derivation of the Kuramoto-Sivashinsky equation from a 2D weakly nonlinear Stefan problem

C.M Brauner, J. Hulshof, L. Lorenzi

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We are interested in a rigorous derivation of the Kuramoto-Sivashinsky (K-S) equation from a free boundary problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter e and define rescaled variables accordingly. At fixed ε, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, and the Lyapunov-Schmidt method. As a solvability condition, we derive a self-consistent parabolic equation for the front. We prove that, starting from the same configuration, the latter remains close to the solution of K-S on a fixed time interval, uniformly in ε\ sufficiently small. © European Mathematical Society 2011.
Original languageEnglish
Article number1
Pages (from-to)73-103
Number of pages30
JournalInterfaces and Free Boundaries
Volume13
DOIs
Publication statusPublished - 2011

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