Electronic Journal of Differential Equations (Sep 2007)
A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom
Abstract
The spatially periodic Kuramoto-Sivashinsky equation (pKS) $$ partial_t u=-partial_x^4 u-c_3partial_x^3 u-c_2partial_x^2 u+2 deltapartial_x(cos( x)u)-partial_x(u^2), $$ with $u(t,x)inmathbb{R}$, $tgeq 0$, $xinmathbb{R}$, is a model problem for inclined film flow over wavy bottoms and other spatially periodic systems with a long wave instability. For given $c_2,c_3inmathbb{R}$ and small $deltageq 0$ 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 $u_m=frac 1 {2pi}int_0^{2pi} u_s(x) ,{ m d} x$. 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.
