We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or weakly unstable limit $Fto 2^+$ and for general values of the Froude number $
F$, including the limit $Fto +infty$. In the former, $Fto 2^+$, 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 this formal limit, showing that stability as $Fto 2^+$ is equivalent to stability of the corresponding KdV-KS waves 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 remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around $F=2.3$ from weakly unstable to different, large-$F$ behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $2.5leq Fleq 6.0$.
Partial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these wave
s is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability -in fact, emph{orbital} stability, since we are dealing with space-invariant problems-, is far from being straightforward when the best spectral stability we can expect is a emph{neutral} one. Indeed, because of the Hamiltonian structure, the spectrum of the linearized equations cannot be bounded away from the imaginary axis, even if we manage to deal with the point zero, which is always present because of space invariance. Some other means make a crucial use of the underlying structure. This is clearly the case for the variational approach, which basically uses the Hamiltonian -or more precisely, a constrained functional associated with the Hamiltonian and with other conserved quantities- as a Lyapunov function. When it works, it is very powerful, since it gives a straight path to orbital stability. An alternative is the modulational approach, following the ideas developed by Whitham almost fifty years ago. The main purpose here is to point out a few results, for KdV-like equations and systems, that make the connection between these three approaches: spectral, variational, and modulational.
Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully non
linear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models are considered, and a most basic question is investigated, namely the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg--de Vries equation, and the less known Euler--Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water waves theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre [Comm. Partial Differential Equations 2005], Oh and Zumbrun [Arch. Ration. Mech. Anal. 2003], and Johnson, Zumbrun and Bronski [Phys. D 2010]. In addition, reduced necessary conditions are obtained in the small amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler--Korteweg system with two types of pressure laws, namely the quadratic law of shallow water equations, and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.
In this paper, we derive consistent shallow water equations for bi-layer flows of Newtonian fluids flowing down a ramp. We carry out a complete spectral analysis of steady flows in the low frequency regime and show the occurence of hydrodynamic insta
bilities, so called roll-waves, when steady flows are unstable.
This paper is concerned with the stability of periodic wave trains in a generalized Kuramoto-Sivashinski (gKS) equation. This equation is useful to describe the weak instability of low frequency perturbations for thin film flows down an inclined ramp
. We provide a set of equations, namely Whithams modulation equations, that determines the behaviour of low frequency perturbations of periodic wave trains. As a byproduct, we relate the spectral stability in the small wavenumber regime to properties of the modulation equations. This stability is always critical since 0 is a 0-Floquet number eigenvalue associated to translational invariance.
This paper is concerned with the detailed behaviour of roll-waves undergoing a low-frequency perturbation. We rst derive the so-called Whithams averaged modulation equations and relate the well-posedness of this set of equations to the spectral stabi
lity problem in the small Floquet-number limit. We then fully validate such a system and in particular, we are able to construct solutions to the shallow water equations in the neighbourhood of modulated roll-waves proles that exist for asymptotically large time.
The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of equations is similar to compr
essible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier-Stokes equations with constant viscosity and free surface. We prove that for a set of suitable initial data (asymptotically close to shallow water initial data), the Cauchy problem for these equations is well-posed, and the solution converges to the solution of viscous shallow water equations. More precisely, we build the solution of the full problem as a perturbation of the strong solution to the viscous shallow water equations. The method of proof is based on a Lagrangian change of variable that fixes the fluid domain and we have to prove the well-posedness in thin domains: we have to pay a special attention to constants in classical Sobolev inequalities and regularity in Stokes problem.
