A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom
| dc.contributor.author | Uecker, Hannes | |
| dc.contributor.author | Wierschem, Andreas | |
| dc.date.accessioned | 2021-08-17T14:41:36Z | |
| dc.date.available | 2021-08-17T14:41:36Z | |
| dc.date.issued | 2007-09-06 | |
| dc.description.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<sub>m</sub> = 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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 18 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Uecker, H., & Wierschem, A. (2007). A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom. <i>Electronic Journal of Differential Equations, 2007</i>(118), pp. 1-18. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14333 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University-San Marcos, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2007, San Marcos, Texas: Texas State University-San Marcos and University of North Texas. | |
| dc.subject | Inclined film flow | |
| dc.subject | Wavy bottom | |
| dc.subject | Burgers equation | |
| dc.subject | Stability | |
| dc.subject | Renormalization | |
| dc.title | A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom | |
| dc.type | Article |