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.
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Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Rigorous derivation of the Kuramoto–Sivashinsky equation from a 2D weakly nonlinear Stefan problem. Interfaces Free Bound. 13 (2011), no. 1, pp. 73–103