Geodesic
Rigorous derivation of the Kuramoto–Sivashinsky equation from a 2D weakly nonlinear Stefan problem
Claude-Michel Brauner1 ; Josephus Hulshof2 ; Luca Lorenzi3
1Université de Bordeaux I, Talence, France
2Vrije Universiteit, Amsterdam, Netherlands
3Università di Parma, Italy
Interfaces and free boundaries, Tome 13 (2011) no. 1, pp. 73-103

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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 ε 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.
DOI : 10.4171/ifb/249
Classification : 35-XX, 80-XX, 00-XX
Mots-clés : Kuramoto–Sivashinsky equation; front dynamics; Stefan problem; singular perturbations; pseudo-differential operators

Claude-Michel Brauner  1   ; Josephus Hulshof  2   ; Luca Lorenzi  3

1 Université de Bordeaux I, Talence, France
2 Vrije Universiteit, Amsterdam, Netherlands
3 Università di Parma, Italy 
Claude-Michel Brauner; Josephus Hulshof; Luca Lorenzi. Rigorous derivation of the Kuramoto–Sivashinsky equation from a 2D weakly nonlinear Stefan problem. Interfaces and free boundaries, Tome 13 (2011) no. 1, pp. 73-103. doi: 10.4171/ifb/249
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     title = {Rigorous derivation of the {Kuramoto{\textendash}Sivashinsky} equation from a {2D} weakly nonlinear {Stefan} problem},
     journal = {Interfaces and free boundaries},
     pages = {73--103},
     year = {2011},
     volume = {13},
     number = {1},
     doi = {10.4171/ifb/249},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/249/}
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