52 research outputs found
Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions
Under natural spectral stability assumptions motivated by previous
investigations of the associated spectral stability problem, we determine sharp
estimates on the linearized solution operator about a multidimensional
planar periodic wave of a system of conservation laws with viscosity, yielding
linearized stability for all and dimensions and nonlinear stability and
-asymptotic behavior for and . The behavior can in
general be rather complicated, involving both convective (i.e., wave-like) and
diffusive effects
Nonlinear Stability of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Dimensions One and Two
This is the published version, also available here: http://dx.doi.org/10.1137/100781808.Extending results of Oh and Zumbrun in dimensions , we establish nonlinear stability and asymptotic behavior of spatially periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions , under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the nonlinear iteration in the Duhamel representation of a modulated periodic wave
Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
International audienceWe study the spectral stability of roll-wave solutions of the viscous St. Venant equationsmodeling inclined shallow-water flow, both at onset in the small-Froude number or “weakly unstable”limit F → 2+ and for general values of the Froude number F , including the limit F → +∞. In the former,F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV)equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate thisformal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KSwaves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St.Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainderof the paper, we investigate numerically and analytically the evolution of the stability diagram as Froudenumber increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable todifferent, large-F behavior, with stability determined by simple power law relations. The latter stabilitycriteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0
[Book of abstracts]
USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos
- …