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 |