Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

Baldi, Pietro

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 33-77 / Harvested from Numdam

Résumé

Nous démontrons lʼexistence de solutions de petite amplitude, périodiques en temps, de versions quasi-linéaires ou complètement non linéaires de lʼéquation de Benjamin–Ono, dans le cas complètement résonnant. Le résultat est démontré dans lʼéchelle de Sobolev, pour des fréquences dans un ensemble de Cantor de mesure asymptotiquement pleine à lʼorigine. Nous considérons le cas général où lʼéquation peut-être vue comme un système Hamiltonien réversible perturbé par une partie réversible, mais qui nʼest pas Hamiltonienne, contenant des dérivées dʼordre maximal.Ces solutions sont, au premier ordre, obtenues par superposition dʼun nombre arbitrairement grand dʼondes se propageant à des vitesses différentes (solutions multimodales).Les principales difficultés du problème sont : la présence dʼun noyau de dimension infinie pour lʼéquation de bifurcation, et lʼoccurence de petits diviseurs dans la résolution de lʼéquation linéarisée, où les dérivées de plus haut degré ont des coefficients variables.Nous montrons que lʼopérateur linéarisé est essentiellement conjugué à un opérateur à coefficients constants, modulo un terme régularisant. La démonstration, basée sur des changements de variables et des conjugaisons avec des opérateurs pseudo-différentiels, est inspirée par la méthode utilisée par Iooss, Plotnikov et Toland (ARMA 2005) pour démontrer lʼexistence dʼondes de gravité stationnaires. La démonstration utilise également un schéma de Nash–Moser adapté à ce contexte, dans lʼéchelle des espaces de Sobolev, ainsi quʼune décomposition de Lyapunov–Schmidt.

Abstract

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions).The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order.The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have non-constant coefficients.The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash–Moser iteration in Sobolev spaces, and Lyapunov–Schmidt decomposition.

Détail

Publié le : 2013-01-01

MR 3011291 | Zbl 1285.35090

DOI : https://doi.org/10.1016/j.anihpc.2012.06.001
Classification:  35B10,  37K55,  37K50

Bibliographie

[1] M. Ablowitz, A. Fokas, The inverse scattering transform for the Benjamin–Ono equation, a pivot for multidimensional problems, Stud. Appl. Math. 68 (1983), 1-10 | MR 686248 | Zbl 0505.76031

[2] D.M. Ambrose, J. Wilkening, Computation of time-periodic solutions of the Benjamin–Ono equation, J. Nonlinear Sci. 20 no. 3 (2010), 277-308 | MR 2639896 | Zbl 1203.37085

[3] P. Baldi, Periodic solutions of forced Kirchhoff equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), 117-141 | Numdam | MR 2512203 | Zbl 1180.35040

[4] D. Bambusi, S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Commun. Math. Phys. 219 (2001), 465-480 | MR 1833810 | Zbl 1003.37042

[5] T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-562 | Zbl 0147.46502

[6] M. Berti, Nonlinear Oscillations of Hamiltonian PDEs, Progress in Nonlin. Diff. Eq. Appl. vol. 74, Birkhäuser, Boston (2008) | MR 2345400

[7] M. Berti, L. Biasco, M. Procesi, KAM theory for the Hamiltonian derivative wave equation, arXiv:1111.3905 (2011) | MR 3112201

[8] M. Berti, Ph. Bolle, Cantor families of periodic solutions of wave equations with $ {C}^{k}$ nonlinearities, NoDEA 15 (2008), 247-276 | MR 2408354 | Zbl 1155.35004

[9] M. Berti, Ph. Bolle, M. Procesi, An abstract Nash–Moser theorem with parameters and applications to PDEs, Ann. Institute H. Poincaré (C) Anal. Non Linéaire 27 (2010), 377-399 | Numdam | MR 2580515 | Zbl 1203.47038

[10] J. Bourgain, Periodic solutions of nonlinear wave equations, Harmonic Analysis and Partial Differential Equations, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL (1999), 69-97 | MR 1743856 | Zbl 0976.35041

[11] N. Burq, F. Planchon, On well-posedness for the Benjamin–Ono equation, Math. Ann. 340 no. 3 (2005), 497-542 | MR 2357995 | Zbl 1148.35074

[12] J. Colliander, C. Kenig, G. Staffilani, Local well-posedness for dispersion generalized Benjamin–Ono equations, Diff. Integral Eq. Internat. J. Theory Appl. 16 no. 12 (2003), 1441-1472 | MR 2029909 | Zbl 1075.35548

[13] W. Craig, Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas et Synthèses vol. 9, Société Mathématique de France, Paris (2000) | MR 1804420

[14] J.-M. Delort, A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein–Gordon equation on $ {\mathbb{S}}^{1}$, Astérisque 341 (2012) | MR 2952065

[15] R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 no. 1 (1982), 65-222 | MR 656198 | Zbl 0499.58003

[16] L. Hörmander, The boundary problems of physical geodesy, Arch. Ration. Mech. Anal. 62 no. 1 (1976), 1-52 | MR 602181 | Zbl 0331.35020

[17] H. Inci, T. Kappeler, P. Topalov, On the regularity of the composition of diffeomorphisms, Mem. Amer. Math. Soc., in press. | MR 3135704

[18] A.D. Ionescu, C.E. Kenig, Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), 753-798 | MR 2291918 | Zbl 1123.35055

[19] G. Iooss, P.I. Plotnikov, Multimodal standing gravity waves: a completely resonant system, J. Math. Fluid Mech. 7 (2005), 110-126 | MR 2126133 | Zbl 1065.35051

[20] G. Iooss, P.I. Plotnikov, Existence of multimodal standing gravity waves, J. Math. Fluid Mech. 7 (2005), 349-364 | MR 2168247 | Zbl 1096.76009

[21] G. Iooss, P.I. Plotnikov, J.F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal. 177 no. 3 (2005), 367-478 | MR 2187619 | Zbl 1176.76017

[22] T. Kappeler, J. Pöschel, KdV & KAM, Springer-Verlag (2003) | MR 1997070 | Zbl 1032.37001

[23] S. Klainerman, A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math. 33 (1980), 241-263 | MR 562736 | Zbl 0443.35040

[24] P. Koosis, Introduction to $ {H}^{p}$ Spaces, Cambridge University Press (1998) | MR 1669574

[25] S. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lectures Ser. Math. Appl. vol. 19, Oxford University Press (2000) | MR 1857574

[26] J. Liu, X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys. 307 no. 3 (2011), 629-673 | MR 2842962 | Zbl 1247.37082

[27] L. Molinet, Global well-posedness in the energy space for the Benjamin–Ono equation on the circle, Math. Ann. 337 no. 2 (2007), 353-383 | MR 2262788 | Zbl 1140.35001

[28] L. Molinet, Global well-posedness in $ {L}^{2}$ for the periodic Benjamin–Ono equation, Amer. J. Math. 130 no. 3 (2008), 635-683 | MR 2418924 | Zbl 1157.35001

[29] L. Molinet, J.C. Saut, N. Tzvetkov, Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal. 33 (2001), 982-988 | MR 1885293 | Zbl 0999.35085

[30] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136-176 | MR 208078 | Zbl 0149.29903

[31] J. Moser, A rapidly convergent iteration method and non-linear partial differential equations – I, Ann. Sc. Norm. Super. Pisa Cl. Sci. III Ser. 20 no. 2 (1966), 265-315 | Numdam | MR 199523 | Zbl 0144.18202

[32] P.I. Plotnikov, J.F. Toland, Nash–Moser theory for standing water waves, Arch. Ration. Mech. Anal. 159 (2001), 1-83 | MR 1854060 | Zbl 1033.76005

[33] P.H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967), 145-205 | MR 206507 | Zbl 0152.10003

[34] P.H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations. II, Comm. Pure Appl. Math. 22 (1969), 15-39 | MR 236504 | Zbl 0157.17301

[35] T. Tao, Global well-posedness of the Benjamin–Ono equation in $ {H}^{1}(\mathbb{R})$, J. Hyperbolic Diff. Eq. 1 (2004), 27-49 | MR 2052470 | Zbl 1055.35104

[36] J. Zhang, M. Gao, X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity 24 no. 4 (2011), 1189-1229 | MR 2776117 | Zbl 1216.37025