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) ∂tu = -∂4xu - c3∂3xu - c2∂2xu + 2δ∂x (cos(x)u) - ∂x(u2), with u(t, x) ∈ ℝ, t ≥ 0, x ∈ ℝ, is a model problem for inclined film flow over wavy bottoms and other spatially periodic systems with a long wave instability. For given c2, c3 ∈ ℝ and small δ ≥ 0 it has a one dimensional family of spatially periodic stationary solutions us (⋅; c2, c3, δ, um), parameterized by the mass u_m = 1/2π ∫2π0 us(x) dx. 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 non-linear self-similar way: the limiting profile is determined by a Burgers equation in Bloch wave space. We also discuss linearly unstable us, 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.