Space-modulated stability and averaged dynamics
Journées équations aux dérivées partielles (2015), article no. 8, 15 p.

In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg–de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.

DOI : https://doi.org/10.5802/jedp.637
Classification : 35B10,  35B35,  35K59,  35P05,  35Q53,  37K45
Mots clés : periodic traveling waves, stability, modulation
@article{JEDP_2015____A8_0,
author = {Rodrigues, Luis Miguel},
title = {Space-modulated stability and averaged dynamics},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {8},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2015},
doi = {10.5802/jedp.637},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.637/}
}
TY  - JOUR
AU  - Rodrigues, Luis Miguel
TI  - Space-modulated stability and averaged dynamics
JO  - Journées équations aux dérivées partielles
PY  - 2015
DA  - 2015///
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.637/
UR  - https://doi.org/10.5802/jedp.637
DO  - 10.5802/jedp.637
LA  - en
ID  - JEDP_2015____A8_0
ER  - 
Rodrigues, Luis Miguel. Space-modulated stability and averaged dynamics. Journées équations aux dérivées partielles (2015), article  no. 8, 15 p. doi : 10.5802/jedp.637. http://www.numdam.org/articles/10.5802/jedp.637/

[1] Angulo Pava, Jaime Nonlinear dispersive equations, Mathematical Surveys and Monographs, 156, American Mathematical Society, Providence, RI, 2009, pp. xii+256 (Existence and stability of solitary and periodic travelling wave solutions) | MR 2567568 | Zbl 1202.35246

[2] Barker, Blake Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, J. Differential Equations, Volume 257 (2014) no. 8, pp. 2950-2983 | Article | MR 3249277 | Zbl 1300.35121

[3] Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin Note on the stability of viscous roll-waves (Submitted)

[4] Barker, Blake; Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin Stability of St. Venant roll-waves: from onset to the large-Froude number limit (Submitted)

[5] Benzoni-Gavage, Sylvie; Mietka, Colin; Rodrigues, L. Miguel Co-periodic stability of periodic waves in some Hamiltonian PDEs (Submitted)

[6] Benzoni-Gavage, Sylvie; Noble, Pascal; Rodrigues, L. Miguel Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids, J. Nonlinear Sci., Volume 24 (2014) no. 4, pp. 711-768 | Article | MR 3228473

[7] Bottman, Nate; Deconinck, Bernard KdV cnoidal waves are spectrally stable, Discrete Contin. Dyn. Syst., Volume 25 (2009) no. 4, pp. 1163-1180 | Article | MR 2552133 | Zbl 1178.35327

[8] Gardner, Robert A. Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math., Volume 491 (1997), pp. 149-181 | Article | MR 1476091 | Zbl 0883.35055

[9] Gohberg, Israel; Goldberg, Seymour; Kaashoek, Marinus A. Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, 49, Birkhäuser Verlag, Basel, 1990, pp. xiv+468 | Article | MR 1130394 | Zbl 0745.47002

[10] Henry, Daniel Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981, pp. iv+348 | MR 610244 | Zbl 0456.35001

[11] Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 2, pp. 693-715 | MR 3005327 | Zbl 1276.35031

[12] Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation, Arch. Ration. Mech. Anal., Volume 207 (2013) no. 2, pp. 669-692 | MR 3005326 | Zbl 1270.35106

[13] Johnson, Mathew A.; Noble, Pascal; Rodrigues, L. Miguel; Zumbrun, Kevin Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations, Invent. Math., Volume 197 (2014) no. 1, pp. 115-213 | Article | MR 3219516 | Zbl 1304.35192

[14] Johnson, Mathew A.; Zumbrun, Kevin Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case, J. Differential Equations, Volume 249 (2010) no. 5, pp. 1213-1240 | MR 2652171 | Zbl 1198.35027

[15] Johnson, Mathew A.; Zumbrun, Kevin Nonlinear stability of periodic traveling-wave solutions of viscous conservation laws in dimensions one and two, SIAM J. Appl. Dyn. Syst., Volume 10 (2011) no. 1, pp. 189-211 | MR 2788923 | Zbl 1221.35055

[16] Jung, Soyeun Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves, J. Differential Equations, Volume 253 (2012) no. 6, pp. 1807-1861 | Article | MR 2943944 | Zbl 1268.35015

[17] Jung, Soyeun Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws, J. Differential Equations, Volume 256 (2014) no. 7, pp. 2261-2306 | Article | MR 3160443 | Zbl 1288.35078

[18] Kabil, Buğra; Rodrigues, L. Miguel Spectral validation of the Whitham equations for periodic waves of lattice dynamical systems, J. Differential Equations, Volume 260 (2016) no. 3, pp. 2994-3028 | Article | MR 3427688

[19] Kapitula, Todd; Promislow, Keith Spectral and dynamical stability of nonlinear waves, Applied Mathematical Sciences, 185, Springer, New York, 2013, pp. xiv+361 (With a foreword by Christopher K. R. T. Jones) | Article | MR 3100266 | Zbl 1297.37001

[20] Keldyš, M. V. On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Doklady Akad. Nauk SSSR (N.S.), Volume 77 (1951), pp. 11-14 | MR 41353

[21] Keldyš, M. V. The completeness of eigenfunctions of certain classes of nonselfadjoint linear operators, Uspehi Mat. Nauk, Volume 26 (1971) no. 4(160), pp. 15-41 | MR 300125 | Zbl 0225.47008

[22] Liu, Tai-Ping; Zeng, Yanni Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., Volume 125 (1997) no. 599, pp. viii+120 | MR 1357824 | Zbl 0884.35073

[23] Markus, A. S. Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, 71, American Mathematical Society, Providence, RI, 1988, pp. iv+250 (Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver, With an appendix by M. V. Keldyš) | MR 971506 | Zbl 0678.47005

[24] van Neerven, Jan The asymptotic behaviour of semigroups of linear operators, Operator Theory: Advances and Applications, 88, Birkhäuser Verlag, Basel, 1996, pp. xii+237 | Article | MR 1409370 | Zbl 0905.47001

[25] Noble, Pascal; Rodrigues, L. Miguel Whitham’s modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Indiana Univ. Math. J., Volume 62 (2013) no. 3, pp. 753-783 | Article | MR 3164843 | Zbl 1296.35161

[26] Oh, Myunghyun; Zumbrun, Kevin Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions, Z. Anal. Anwend., Volume 25 (2006) no. 1, pp. 1-21 | MR 2215999 | Zbl 1099.35014

[27] Oh, Myunghyun; Zumbrun, Kevin Erratum to: Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 21-23 | MR 2601068 | Zbl 1197.35075

[28] Oh, Myunghyun; Zumbrun, Kevin Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 1-20 | MR 2601067 | Zbl 1197.35075

[29] Rodrigues, L. Miguel Linear asymptotic stability and modulation behavior near periodic waves of the Korteweg–de Vries equation (forthcoming)

[30] Rodrigues, L. Miguel Vortex-like finite-energy asymptotic profiles for isentropic compressible flows, Indiana Univ. Math. J., Volume 58 (2009) no. 4, pp. 1747-1776 | Article | MR 2542978 | Zbl 1170.76046

[31] Rodrigues, L. Miguel Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows (2013) (Habilitation à Diriger des Recherches)

[32] Rodrigues, L. Miguel; Zumbrun, Kevin Periodic-Coefficient Damping Estimates, and Stability of Large-Amplitude Roll Waves in Inclined Thin Film Flow, SIAM J. Math. Anal., Volume 48 (2016) no. 1, pp. 268-280 | Article

[33] Sandstede, Björn; Scheel, Arnd; Schneider, Guido; Uecker, Hannes Diffusive mixing of periodic wave trains in reaction-diffusion systems, J. Differential Equations, Volume 252 (2012) no. 5, pp. 3541-3574 | Article | MR 2876664 | Zbl 1298.35108

[34] Schneider, Guido Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation, Comm. Math. Phys., Volume 178 (1996) no. 3, pp. 679-702 | MR 1395210 | Zbl 0861.35107

[35] Schneider, Guido 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) (Tohoku Math. Publ.), Volume 8 (1998), pp. 159-167 | MR 1617491 | Zbl 0907.35015

[36] Serre, Denis Spectral stability of periodic solutions of viscous conservation laws: large wavelength analysis, Comm. Partial Differential Equations, Volume 30 (2005) no. 1-3, pp. 259-282 | Article | MR 2131054 | Zbl 1131.35046

[37] Whitham, Gerald B. Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York, 1974, pp. xvi+636 (Pure and Applied Mathematics) | MR 483954 | Zbl 0940.76002

[38] Yakubov, Sasun Completeness of root functions of regular differential operators, Pitman Monographs and Surveys in Pure and Applied Mathematics, 71, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994, pp. x+245 | MR 1401350 | Zbl 0833.34081

Cité par Sources :