Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit
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- by Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues and Kevin Zumbrun PDF
- Trans. Amer. Math. Soc. 367 (2015), 2159-2212 Request permission
Abstract:
We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation $\partial _t v+v\partial _x v+\partial _x^3 v+\delta \left (\partial _x^2 v +\partial _x^4 v\right )=0,$ $\delta >0$, in the Korteweg-de Vries limit $\delta \to 0$, a canonical limit describing small-amplitude weakly unstable thin film flow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions, each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non-empty set of parameter space. The main technical difficulty in our analysis, and one that has not been previously addressed by any authors, is that of obtaining a useful description for $0<\delta \ll 1$ of the spectrum of the associated linearized operators in a sufficiently small neighborhood of the origin in the spectral plane. This modulational stability analysis is particularly interesting, relying on direct calculations of a reduced periodic Evans function and using in an essential way an analogy with hyperbolic relaxation theory at the level of the associated Whitham modulation equations. A second technical difficulty is the exclusion of high-frequency instabilities lying between the $\mathcal {O}(1)$ regime treatable by classical perturbation methods and the $\gtrsim \delta ^{-1}$ regime excluded by parabolic energy estimates.References
- Doron E. Bar and Alexander A. Nepomnyashchy, Stability of periodic waves governed by the modified Kawahara equation, Phys. D 86 (1995), no. 4, 586–602. MR 1353179, DOI 10.1016/0167-2789(95)00174-3
- Blake Barker, Jeffrey Humpherys, and Kevin Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, J. Differential Equations 249 (2010), no. 9, 2175–2213. MR 2718655, DOI 10.1016/j.jde.2010.07.019
- Nate Bottman and Bernard Deconinck, KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst. 25 (2009), no. 4, 1163–1180. MR 2552133, DOI 10.3934/dcds.2009.25.1163
- B. Barker, Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, to appear, J. Diff. Eq.
- Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun, Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation, Phys. D 258 (2013), 11–46. MR 3079606, DOI 10.1016/j.physd.2013.04.011
- Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, and Kevin Zumbrun, Stability of periodic Kuramoto-Sivashinsky waves, Appl. Math. Lett. 25 (2012), no. 5, 824–829. MR 2888080, DOI 10.1016/j.aml.2011.10.026
- B. Barker, M. A. Johnson, P. Noble, L. M. Rodrigues, and K. Zumbrun, Whitham averaged equations and modulational stability of periodic solutions of hyperbolic-parabolic balance laws. Proceedings and seminars, Centre de Mathématiques de l’École Polytechnique; Conference proceedings, “Journées équations aux dérivées partielles”, 2010, Port d’Albret, France. Available online at: http://jedp.cedram.org/jedp-bin/fitem?id=JEDP_2010___.
- Blake Barker, Mathew A. Johnson, L. Miguel Rodrigues, and Kevin Zumbrun, Metastability of solitary roll wave solutions of the St. Venant equations with viscosity, Phys. D 240 (2011), no. 16, 1289–1310. MR 2813829, DOI 10.1016/j.physd.2011.04.022
- Jared C. Bronski and Mathew A. Johnson, The modulational instability for a generalized Korteweg-de Vries equation, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 357–400. MR 2660515, DOI 10.1007/s00205-009-0270-5
- Jared C. Bronski, Mathew A. Johnson, and Todd Kapitula, An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 6, 1141–1173. MR 2855892, DOI 10.1017/S0308210510001216
- Hsueh-Chia Chang and Evgeny A. Demekhin, Complex wave dynamics on thin films, Studies in Interface Science, vol. 14, Elsevier Science B.V., Amsterdam, 2002. MR 2362445
- H. C. Chang, E. A. Demekhin, and D. I. Kopelevich, Laminarizing effects of dispersion in an active-dissipative nonlinear medium, Phys. D 63 (1993), 299–320.
- Bernard Deconinck and Todd Kapitula, The orbital stability of the cnoidal waves of the Korteweg-de Vries equation, Phys. Lett. A 374 (2010), no. 39, 4018–4022. MR 2683991, DOI 10.1016/j.physleta.2010.08.007
- Arjen Doelman, Björn Sandstede, Arnd Scheel, and Guido Schneider, The dynamics of modulated wave trains, Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105. MR 2507940, DOI 10.1090/memo/0934
- N. M. Ercolani, D. W. McLaughlin, and H. Roitner, Attractors and transients for a perturbed periodic KdV equation: a nonlinear spectral analysis, J. Nonlinear Sci. 3 (1993), no. 4, 477–539. MR 1246704, DOI 10.1007/BF02429875
- H. Freistühler and P. Szmolyan, Spectral stability of small shock waves, Arch. Ration. Mech. Anal. 164 (2002), no. 4, 287–309. MR 1933630, DOI 10.1007/s00205-002-0215-8
- Uriel Frisch, Zhen-Su She, and Olivier Thual, Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model, J. Fluid Mech. 168 (1986), 221–240. MR 861201, DOI 10.1017/S0022112086000356
- R. A. Gardner, On the structure of the spectra of periodic travelling waves, J. Math. Pures Appl. (9) 72 (1993), no. 5, 415–439. MR 1239098
- Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI 10.1007/BF00276840
- Jonathan Goodman, Remarks on the stability of viscous shock waves, Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990) SIAM, Philadelphia, PA, 1991, pp. 66–72. MR 1142641
- Jeffrey Humpherys, Olivier Lafitte, and Kevin Zumbrun, Stability of isentropic Navier-Stokes shocks in the high-Mach number limit, Comm. Math. Phys. 293 (2010), no. 1, 1–36. MR 2563797, DOI 10.1007/s00220-009-0885-2
- Mathew A. Johnson, Kevin Zumbrun, and Jared C. Bronski, On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions, Phys. D 239 (2010), no. 23-24, 2057–2065. MR 2733113, DOI 10.1016/j.physd.2010.07.012
- M. A. Johnson, P. Noble, L. M. Rodrigues, and K. Zumbrun, Behaviour of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Invent. Math. 197 (2014), no. 1, 115–213.
- M. A. Johnson and K. Zumbrun, Rigorous justification of the Whitham modulation equations for the generalized Korteweg-de Vries equation, Stud. Appl. Math. 125 (2010), no. 1, 69–89. MR 2676781, DOI 10.1111/j.1467-9590.2010.00482.x
- Mathew A. Johnson and Kevin Zumbrun, Transverse instability of periodic traveling waves in the generalized Kadomtsev-Petviashvili equation, SIAM J. Math. Anal. 42 (2010), no. 6, 2681–2702. MR 2733265, DOI 10.1137/090770758
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432, DOI 10.1007/978-3-642-69689-3
- Y. Kuramoto and T. Tsuzuki. On the formation of dissipative structures in reaction-diffusion systems. Progr. Theoret. Phys., 1975. 54:3.
- E. A. Kuznetsov, M. D. Spector, and G. E. Fal′kovich, On the stability of nonlinear waves in integrable models, Phys. D 10 (1984), no. 3, 379–386. MR 763479, DOI 10.1016/0167-2789(84)90186-6
- Takaaki Nishida, Yoshiaki Teramoto, and Hideaki Yoshihara, Hopf bifurcation in viscous incompressible flow down an inclined plane, J. Math. Fluid Mech. 7 (2005), no. 1, 29–71. MR 2127741, DOI 10.1007/s00021-004-0104-z
- Corrado Mascia and Kevin Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity, Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–263. MR 2004135, DOI 10.1007/s00205-003-0258-5
- Daniel Michelson, Stability of the Bunsen flame profiles in the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 27 (1996), no. 3, 765–781. MR 1382832, DOI 10.1137/0527041
- P. Noble and L. M. Rodrigues, Whitham’s equations for modulated roll-waves in shallow flows, ArXiv e-prints, arXiv:1011.2296v1, 2010.
- Pascal Noble and L. Miguel Rodrigues, Whitham’s modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Indiana Univ. Math. J. 62 (2013), no. 3, 753–783. MR 3164843, DOI 10.1512/iumj.2013.62.4955
- M. Oh and K. Zumbrun, Stability of periodic solutions of conservation laws with viscosity: analysis of the Evans function, Arch. Ration. Mech. Anal. 166 (2003), no. 2, 99–166. MR 1957127, DOI 10.1007/s00205-002-0216-7
- Robert L. Pego, Guido Schneider, and Hannes Uecker, Long-time persistence of Korteweg-de Vries solitons as transient dynamics in a model of inclined film flow, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 1, 133–146. MR 2359776, DOI 10.1017/S0308210505001113
- Ramon Plaza and Kevin Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst. 10 (2004), no. 4, 885–924. MR 2073940, DOI 10.3934/dcds.2004.10.885
- Denis Serre, Spectral stability of periodic solutions of viscous conservation laws: large wavelength analysis, Comm. Partial Differential Equations 30 (2005), no. 1-3, 259–282. MR 2131054, DOI 10.1081/PDE-200044492
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI 10.14492/hokmj/1381757663
- G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations, Acta Astronaut. 4 (1977), no. 11-12, 1177–1206. MR 0502829, DOI 10.1016/0094-5765(77)90096-0
- G. I. Sivashinsky, Instabilities, Pattern Formation, and Turbulence in Flames, Annu. Rev. Fluid Mech. 15 (1983), 179-199.
- G. I. Sivashinsky and D. M. Michelson, On irregular wavy flow of a liquid down an vertical plane, Progr. Theoret. Phys. 63 (6) (1980), 2112-2114.
- M. D. Spektor, Stability of conoidal [cnoidal] waves in media with positive and negative dispersion, Zh. Èksper. Teoret. Fiz. 94 (1988), no. 1, 186–202 (Russian); English transl., Soviet Phys. JETP 67 (1988), no. 1, 104–112. MR 960877
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- Htay Aung Win, Model equation of surface waves of viscous fluid down an inclined plane, J. Math. Kyoto Univ. 33 (1993), no. 3, 803–824. MR 1239094, DOI 10.1215/kjm/1250519194
- Jun Yu and Yi Yang, Evolution of small periodic disturbances into roll waves in channel flow with internal dissipation, Stud. Appl. Math. 111 (2003), no. 1, 1–27. MR 1985993, DOI 10.1111/1467-9590.t01-2-00225
- W.-A. Yong, Basic properties of hyperbolic relaxation systems, Birkhauser’s Series: Progress in Nonlinear Differential Equations and their Applications (2001), 207 pp.
- Yanni Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225–279. MR 1738119, DOI 10.1007/s002050050188
- Kevin Zumbrun, Stability of detonation profiles in the ZND limit, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 141–182. MR 2781588, DOI 10.1007/s00205-010-0342-6
Additional Information
- Mathew A. Johnson
- Affiliation: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66046
- Email: matjohn@ku.edu
- Pascal Noble
- Affiliation: Institut Camille Jordan, UMR CNRS 5208, Université de Lyon, Université Lyon I, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France
- Address at time of publication: Institut de Mathématiques de Toulouse, UMR CNRS 5219, INSA de Toulouse 135, avenue de Rangueil, 31077 Toulouse Cedex 4, France
- Email: noble@math.univ-lyon1.fr, Pascal.Noble@math.univ-toulouse.fr
- L. Miguel Rodrigues
- Affiliation: Institut Camille Jordan, UMR CNRS 5208, Université de Lyon, Université Lyon 1, 43 bd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France
- Email: rodrigues@math.univ-lyon1.fr
- Kevin Zumbrun
- Affiliation: Department of Mathematics, Indiana University, 831 E. 3rd Street, Bloomington, Indiana 47405
- MR Author ID: 330192
- Email: kzumbrun@indiana.edu
- Received by editor(s): February 28, 2012
- Received by editor(s) in revised form: June 20, 2013
- Published electronically: July 17, 2014
- Additional Notes: The research of the first author was partially supported under NSF grant no. 1211183.
The research of the second author was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01.
The stay of the third author in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01.
The research of the fourth author was partially supported under NSF grant no. DMS-0300487 - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 2159-2212
- MSC (2010): Primary 35B35, 35B10, 35Q53
- DOI: https://doi.org/10.1090/S0002-9947-2014-06274-0
- MathSciNet review: 3286511