In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed “metastability" of the the limiting homoclinic orbits.
Mots clés : Periodic traveling waves, St. Venant equations, Spectral stability, Nonlinear stability.
@article{JEDP_2010____A3_0,
author = {Barker, Blake and Johnson, Mathew A. and Noble, Pascal and Rodrigues, L.Miguel and Zumbrun, Kevin},
title = {Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {3},
pages = {1--24},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2010},
doi = {10.5802/jedp.60},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.60/}
}
TY - JOUR AU - Barker, Blake AU - Johnson, Mathew A. AU - Noble, Pascal AU - Rodrigues, L.Miguel AU - Zumbrun, Kevin TI - Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law JO - Journées équations aux dérivées partielles PY - 2010 SP - 1 EP - 24 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.60/ DO - 10.5802/jedp.60 LA - en ID - JEDP_2010____A3_0 ER -
%0 Journal Article %A Barker, Blake %A Johnson, Mathew A. %A Noble, Pascal %A Rodrigues, L.Miguel %A Zumbrun, Kevin %T Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law %J Journées équations aux dérivées partielles %D 2010 %P 1-24 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.60/ %R 10.5802/jedp.60 %G en %F JEDP_2010____A3_0
Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L.Miguel; Zumbrun, Kevin. Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law. Journées équations aux dérivées partielles (2010), article no. 3, 24 p. doi : 10.5802/jedp.60. http://www.numdam.org/articles/10.5802/jedp.60/
[BM] N.J. Balmforth and S. Mandre, Dynamics of roll waves, J. Fluid Mech. 514 (2004), 1–33. | MR | Zbl
[BJNRZ] B. Barker, M. Johnson, P. Noble, M. Rodrigues, and K. Zumbrun, Spectral stability of periodic viscous roll waves, in preparation.
[BJRZ] B. Baker, M. Johnson, M. Rodrigues, and K. Zumbrun, Metastability of Solitary Roll Wave Solutions of the St. Venant Equations with Viscosity, preprint (2010).
[CuD] C. Curtis and B. Deconick, On the convergence of Hill’s method, Mathematics of computation 79 (2010), 169–187. | MR
[CDKK] J. D. Carter, B. Deconick, F. Kiyak, and J. Nathan Kutz, SpectrUW: a laboratory for the numerical exploration of spectra of linear operators, Mathematics and Computers in Simulation 74 (2007), 370–379. | MR | Zbl
[DK] Bernard Deconinck and J. Nathan Kutz, Computing spectra of linear operators using Hill’s method. J. Comp. Physics 219 (2006), 296-321. | MR | Zbl
[G] R. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl. 72 (1993), 415-439. | MR | Zbl
[He] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer–Verlag, Berlin (1981). | MR | Zbl
[HC] S.-H. Hwang and H.-C. Chang, Turbulent and inertial roll waves in inclined film flow, Phys. Fluids 30 (1987), no. 5, 1259–1268. | MR | Zbl
[JZ1] M. Johnson and K. Zumbrun, Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation, Studies in Applied Mathematics, 125 no. 1 (2010), 69-89. | MR | Zbl
[JZ4] M. Johnson and K. Zumbrun, Nonlinear stability and asymptotic behavior of periodic traveling waves of multidimensional viscous conservation laws in dimensions one and two, preprint (2009).
[JZ2] M. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case, J. Diff. Eq. 249 no. 5 (2010), 1213-1240. | MR
[JZ3] M. Johnson and K. Zumbrun, Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations, preprint (2010). | MR
[JZN] M. Johnson, K. Zumbrun, and P. Noble, Nonlinear stability of viscous roll waves, preprint (2010). | MR
[K] T. Kato, Perturbation theory for linear operators, Springer–Verlag, Berlin Heidelberg (1985). | Zbl
[N1] P. Noble, On the spectral stability of roll waves, Indiana Univ. Math. J. 55 (2006), 795–848. | MR
[N2] P. Noble, Linear stability of viscous roll waves, Comm. Partial Differential Equations 32 no. 10-12 (2007), 1681–1713. | MR | Zbl
[NR] P. Noble and L. M. Rodrigues, Whitham’s equations for modulated roll-waves in shallow flows, preprint (2010).
[OZ1] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws with viscosity- 1. Analysis of the Evans function, Arch. Ration. Mech. Anal. 166 no. 2 (2003), 99–166. | MR | Zbl
[OZ2] M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws with viscosity- Pointwise bounds on the Green function, Arch. Ration. Mech. Anal. 166 no. 2 (2003), 167–196. | MR | Zbl
[OZ3] M. Oh, and K. Zumbrun, Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions, Journal for Analysis and its Applications, 25 (2006), 1–21. | MR | Zbl
[OZ4] M. Oh, and K. Zumbrun, Stability and asymptotic behavior of traveling-wave solutions of viscous conservation laws in several dimensions,Arch. Ration. Mech. Anal., 196 no. 1 (2010), 1–20. Erratum: Arch. Ration. Mech. Anal., 196 no. 1 (2010), 21–23. | MR | Zbl
[Pa] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York-Berlin, (1983) viii+279 pp. ISBN: 0-387-90845-5. | MR | Zbl
[PSU] R. Pego, H. Schneider, and H. Uecker, Long-time persistence of Korteweg-de Vries solitons as transient dynamics in a model of inclined film flow, Proc. Royal Soc. Edinburg 137A (2007) 133–146. | MR | Zbl
[S1] G. Schneider, Nonlinear diffusive stability of spatially periodic solutions– abstract theorem and higher space dimensions, Proceedings of the International Conference on Asymptotics in Nonlinear Diffusive Systems (Sendai, 1997), 159–167, Tohoku Math. Publ., 8, Tohoku Univ., Sendai, 1998. | MR | Zbl
[S2] G. Schneider, Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, (English. English summary) Comm. Math. Phys. 178 no. 3 (1996), 679–702. | MR | Zbl
[S3] G. Schneider, Nonlinear stability of Taylor vortices in infinite cylinders, Arch. Rat. Mech. Anal. 144 no. 2 (1998), 121–200. | MR | Zbl
[Se1] D. Serre, Spectral stability of periodic solutions of viscous conservation laws: Large wavelength analysis, Comm. Partial Differential Equations 30 no. 1-3 (2005), 259–282. | MR | Zbl
[W] G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication. | MR | Zbl
Cité par Sources :
