Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 118, pp. 1-18.
Title: A spatially periodic Kuramoto-Sivashinsky equation as a
model problem for inclined film flow over wavy bottom
Authors: Hannes Uecker (Univ. Stuttgart, Germany)
Andreas Wierschem (Technical University of Munich, Germany)
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
$u_m=\frac 1 {2\pi}\int_0^{2\pi} u_s(x) \,{\rm d} x$.
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 06, 2007.
Math Subject Classifications: 35B40, 35Q53.
Key Words: Inclined film flow; wavy bottom; Burgers equation;
stability; renormalization.