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In this paper we are interested in a rigorous derivation of theKuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm,we consider a two-dimensional Stefan problem in a strip, a simplified versionof a solid-liquid interface model. Near the instability threshold, we introducea small parameter $\varepsilon$ and define rescaled variables accordingly. Atfixed $\varepsilon$, our method is based on: definition of a suitable linear 1Doperator, projection with respect to the longitudinal coordinate only,Lyapunov-Schmidt method. As a solvability condition, we derive aself-consistent parabolic equation for the front. We prove that, starting fromthe same configuration, the latter remains close to the solution of K--S on afixed time interval, uniformly in $\varepsilon$ sufficiently small.

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