Title: On the spectral stablility of roll-waves

Authors: Pascal Noble

Issue: Volume 55 (2006), Issue 2, 795-848

Abstract: In this paper, we study the spectral stability of "roll-waves" occurring in shallow water flows. Roll-waves are periodic, discontinuous travelling waves solutions of the Saint Venant equations. This system describes a shallow water flow downstream to a dam with a small slope. The existence of such profiles is well established due to a classical work of Dressler. The purpose of this article is first to give a rigorous notion of stability in the presence of an infinite number of shocks. After a suitable change of variable, we can formulate a spectral problem in $C^1(\mathbb{R} \setminus \{iL,\; i \in \mathbb{Z}\})$ for a given $L$, the Rankine Hugoniot conditions playing the role of boundary conditions. It is proved analytically that there are no unstable eigenvalues with large moduli when the wavelength is large. We carry out a complete spectral stability analysis in the case of roll-waves in Burgers equations. When the slope of the channel goes to zero in Saint Venant equations, Dressler roll-waves are proved to be spectrally stable.