Space-modulated stability and averaged dynamics
Luis Miguel Rodrigues1
1 Université de Rennes 1 IRMAR, UMR CNRS 6625 263 avenue du General Leclerc F-35042 Rennes Cedex France
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: 10.5802/jedp.637
Classification: 35B10, 35B35, 35K59, 35P05, 35Q53, 37K45
Mots-clés : periodic traveling waves, stability, modulation
Luis Miguel Rodrigues 1

1 Université de Rennes 1 IRMAR, UMR CNRS 6625 263 avenue du General Leclerc F-35042 Rennes Cedex France 
@incollection{JEDP_2015____A8_0,
     author = {Luis Miguel Rodrigues},
     title = {Space-modulated stability and averaged dynamics},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {8},
     pages = {1--15},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2015},
     doi = {10.5802/jedp.637},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.637/}
}
TY  - JOUR
AU  - Luis Miguel Rodrigues
TI  - Space-modulated stability and averaged dynamics
JO  - Journées équations aux dérivées partielles
PY  - 2015
SP  - 1
EP  - 15
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.637/
DO  - 10.5802/jedp.637
LA  - en
ID  - JEDP_2015____A8_0
ER  - 
%0 Journal Article
%A Luis Miguel Rodrigues
%T Space-modulated stability and averaged dynamics
%J Journées équations aux dérivées partielles
%D 2015
%P 1-15
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.637/
%R 10.5802/jedp.637
%G en
%F JEDP_2015____A8_0
Luis Miguel Rodrigues. Space-modulated stability and averaged dynamics. Journées équations aux dérivées partielles (2015), article  no. 8, 15 p. doi : 10.5802/jedp.637. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.637/

[1] Jaime Angulo Pava 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 | Zbl

[2] Blake Barker 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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[14] Mathew A. Johnson; Kevin Zumbrun 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 | Zbl

[15] Mathew A. Johnson; Kevin Zumbrun 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 | Zbl

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

[17] Soyeun Jung 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 | DOI | MR | Zbl

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

[19] Todd Kapitula; Keith Promislow 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) | DOI | MR | Zbl

[20] M. V. Keldyš 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

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

[22] Tai-Ping Liu; Yanni Zeng 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 | Zbl

[23] A. S. Markus 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 | Zbl

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

[25] Pascal Noble; 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., Volume 62 (2013) no. 3, pp. 753-783 | DOI | MR | Zbl

[26] Myunghyun Oh; Kevin Zumbrun 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 | Zbl

[27] Myunghyun Oh; Kevin Zumbrun 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 | Zbl

[28] Myunghyun Oh; Kevin Zumbrun 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 | Zbl

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

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

[31] L. Miguel Rodrigues Asymptotic stability and modulation of periodic wavetrains, general theory & applications to thin film flows, Université Lyon 1 (2013) (Habilitation à Diriger des Recherches)

[32] L. Miguel Rodrigues; Kevin Zumbrun 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 | DOI

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

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

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

[36] Denis Serre 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 | DOI | MR | Zbl

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

[38] Sasun Yakubov 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 | Zbl

Cited by Sources: