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Asymptotic stability of steady-states for Saint-Venant equations with real viscosity. (English) Zbl 1291.76053

Calgaro, Catarina (ed.) et al., Analysis and simulation of fluid dynamics. Collected papers based on the presentations at the conference, Lille, France, June 2005. Basel: Birkhäuser (ISBN 3-7643-7741-0/hbk). Advances in Mathematical Fluid Mechanics, 155-162 (2007).
Introduction: The Saint-Venant model, introduced in [A.-J.-C. B. de Saint-Venant, C. R. Acad. Sci. Paris 73, 147–154, 237–240 (1871; JFM 03.0482.04)], gives rise to a simple and, nevertheless, rich hyperbolic system of conservation laws with a zero-order reaction term. Apart from its interest as a model itself, we consider the analysis of the Saint-Venant system a useful preliminary step in the understanding of the role played by lower-order terms in qualitative properties/asymptotic behavior of solutions to conservation laws. In what follows we deal with the viscous Saint-Venant model and analyze the stability of stationary steady states. We skip all of the technical details of the analysis, stressing only the kind of result we are able to prove at the present and the main lines of the proof. Details will be given in a forthcoming paper.
For the entire collection see [Zbl 1106.76005].

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B35 Stability in context of PDEs
35Q35 PDEs in connection with fluid mechanics
76E09 Stability and instability of nonparallel flows in hydrodynamic stability

Citations:

JFM 03.0482.04
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References:

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