Uecker H, Wierschem A (2007)
Publication Type: Journal article, Original article
Publication year: 2007
Publisher: Texas State University, Department of Mathematics
Book Volume: 2007
Pages Range: 1-18
Journal Issue: 118
URI: http://ejde.math.txstate.edu/Volumes/2007/118/uecker.pdf
The spatially periodic Kuramoto-Sivashinsky equation (pKS) ∂tu = - ∂x4u - c3∂ x3u - c2∂x2u + 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 um = 1/2π ∫02π 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. © 2007 Texas State University - San Marcos.
APA:
Uecker, H., & Wierschem, A. (2007). A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom. Electronic Journal of Differential Equations, 2007(118), 1-18.
MLA:
Uecker, Hannes, and Andreas Wierschem. "A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom." Electronic Journal of Differential Equations 2007.118 (2007): 1-18.
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