Electron. J. Diff. Eqns., Vol. 2007(2007), No. 118, pp. 1-18.

A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom

Hannes Uecker, Andreas Wierschem

Abstract:
The spatially periodic Kuramoto-Sivashinsky equation (pKS)
$$ \partial_t u=-\partial_x^4 u-c_3\partial_x^3 u-c_2\partial_x^2 u+2 \delta\partial_x(\cos( x)u)-\partial_x(u^2), $$
with $u(t,x)\in\mathbb{R}$, $t\geq 0$, $x\in\mathbb{R}$, is a model problem for inclined film flow over wavy bottoms and other spatially periodic systems with a long wave instability. For given $c_2,c_3\in\mathbb{R}$ and small $\delta\geq 0$ it has a one dimensional family of spatially periodic stationary solutions $u_s(\cdot;c_2,c_3,\delta,u_m)$, parameterized by the mass $u_m=\frac 1 {2\pi}\int_0^{2\pi} u_s(x) \,d x$. 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 $u_s$, 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
Universität Stuttgart
D-70569 Stuttgart, Germany
email: hannes.uecker@mathematik.uni-stuttgart.de
Andreas Wierschem
Fluid Mechanics and Process Automation
Technical University of Munich
D-85350 Freising, Germany
email: wiersche@wzw.tum.de

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