### 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 language | English |
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Article number | 1 |

Pages (from-to) | 73-103 |

Number of pages | 30 |

Journal | Interfaces and Free Boundaries |

Volume | 13 |

DOIs | |

Publication status | Published - 2011 |

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## Cite this

Brauner, C. M., Hulshof, J., & Lorenzi, L. (2011). Rigorous derivation of the Kuramoto-Sivashinsky equation from a 2D weakly nonlinear Stefan problem.

*Interfaces and Free Boundaries*,*13*, 73-103. [1]. https://doi.org/10.4171/IFB/249