Hannes Uecker, Andreas Wierschem
The spatially periodic Kuramoto-Sivashinsky equation (pKS)
with , , , is a model problem for inclined film flow over wavy bottoms and other spatially periodic systems with a long wave instability. For given and small it has a one dimensional family of spatially periodic stationary solutions , parameterized by the mass . Depending on the parameters these stationary solutions can be linearly stable or unstable. We show that in the stable case localized perturbations decay with a polynomial rate and in a universal nonlinear self-similar way: the limiting profile is determined by a Burgers equation in Bloch wave space. We also discuss linearly unstable , in which case we approximate the pKS by a constant coefficient KS-equation. The analysis is based on Bloch wave transform and renormalization group methods.
Submitted May 15, 2007. Published September 6, 2007.
Math Subject Classifications: 35B40, 35Q53.
Key Words: Inclined film flow; wavy bottom; Burgers equation; stability; renormalization.
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| Hannes Uecker |
Institut für Analysis, Dynamik und Modellierung
D-70569 Stuttgart, Germany
| Andreas Wierschem |
Fluid Mechanics and Process Automation
Technical University of Munich
D-85350 Freising, Germany
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