Birkhoff normal forms for Hamiltonian PDEs in their energy space
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 681-745.

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a sufficiently strong energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that only modes with the same linear frequency will be able to exchange energy in a reasonable time. Contrary to the previous existing results, we do not require the solutions to be particularly regular. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d=1 and nonlinear Schrödinger equations in dimension d2.

On étudie le comportement en temps long des petites solutions d’équations dispersives hamiltoniennes semi-linéaires sur des domaines bornés. Si le système satisfait à une nouvelle condition de non-résonance et à une estimée d’énergie suffisamment forte, on prouve que ses basses super-actions sont quasiment préservées pendant des temps très longs. En d’autres termes cela signifie que, pour échanger de l’énergie, les modes doivent osciller à la même fréquence. La nouveauté de ce résultat est que l’on n’a pas à supposer que les solutions sont particulièrement régulières. Il suffit qu’elles soient dans l’espace d’énergie. On applique notre résultat aux équations de Klein-Gordon en dimension d=1 ainsi qu’aux équations de Schrödinger non linéaires en dimension d2.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.193
Classification: 35Q55, 81Q05, 37K55
Keywords: Birkhoff normal forms, dispersive equations, low regularity, Hamiltonian PDE, Sturm–Liouville
Mot clés : Formes normales de Birkhoff, équations dispersives, faible régularité, EDP hamiltonienne, Sturm–Liouville
Joackim Bernier 1; Benoît Grébert 1

1 Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL F-44000 Nantes, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2022__9__681_0,
     author = {Joackim Bernier and Beno{\^\i}t Gr\'ebert},
     title = {Birkhoff normal forms for {Hamiltonian} {PDEs} in their energy space},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {681--745},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.193},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.193/}
}
TY  - JOUR
AU  - Joackim Bernier
AU  - Benoît Grébert
TI  - Birkhoff normal forms for Hamiltonian PDEs in their energy space
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 681
EP  - 745
VL  - 9
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.193/
DO  - 10.5802/jep.193
LA  - en
ID  - JEP_2022__9__681_0
ER  - 
%0 Journal Article
%A Joackim Bernier
%A Benoît Grébert
%T Birkhoff normal forms for Hamiltonian PDEs in their energy space
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 681-745
%V 9
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.193/
%R 10.5802/jep.193
%G en
%F JEP_2022__9__681_0
Joackim Bernier; Benoît Grébert. Birkhoff normal forms for Hamiltonian PDEs in their energy space. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 681-745. doi : 10.5802/jep.193. https://jep.centre-mersenne.org/articles/10.5802/jep.193/

[Agr15] M. S. Agranovich - Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains, Springer Monographs in Math., Springer, Cham, 2015 | DOI | MR

[Bam99] D. Bambusi - “Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations”, Math. Z. 230 (1999) no. 2, p. 345-387 | DOI | Zbl

[Bam03] D. Bambusi - “Birkhoff normal form for some nonlinear PDEs”, Comm. Math. Phys. 234 (2003) no. 2, p. 253-285 | DOI | MR | Zbl

[BB13] M. Berti & P. Bolle - “Quasi-periodic solutions with Sobolev regularity of NLS on 𝕋 d with a multiplicative potential”, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 1, p. 229-286 | DOI | MR | Zbl

[BD18] M. Berti & J.-M. Delort - Almost global solutions of capillary-gravity water waves equations on the circle, Lect. Notes of the Unione Matematica Italiana, vol. 24, Springer, Cham, 2018 | DOI

[BDGS07] D. Bambusi, J.-M. Delort, B. Grébert & J. Szeftel - “Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds”, Comm. Pure Appl. Math. 60 (2007) no. 11, p. 1665-1690 | DOI | MR | Zbl

[BFG20a] J. Bernier, E. Faou & B. Grébert - “Long time behavior of the solutions of NLW on the d-dimensional torus”, Forum Math. Sigma 8 (2020), article ID e12, 26 pages | DOI | MR | Zbl

[BFG20b] J. Bernier, E. Faou & B. Grébert - “Rational normal forms and stability of small solutions to nonlinear Schrödinger equations”, Ann. PDE 6 (2020) no. 2, article ID 14, 65 pages | DOI | Zbl

[BG06] D. Bambusi & B. Grébert - “Birkhoff normal form for partial differential equations with tame modulus”, Duke Math. J. 135 (2006) no. 3, p. 507-567 | DOI | MR | Zbl

[BG21] J. Bernier & B. Grébert - “Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations”, Arch. Rational Mech. Anal. 241 (2021) no. 3, p. 1139-1241 | DOI | MR | Zbl

[BGR21] J. Bernier, B. Grébert & G. Rivière - “Dynamics of nonlinear Klein-Gordon equations in low regularity on 𝕊 2 , 2021 | arXiv

[BGT05] N. Burq, P. Gérard & N. Tzvetkov - “Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations”, Ann. Sci. École Norm. Sup. (4) 38 (2005) no. 2, p. 255-301 | DOI | Numdam | Zbl

[BMP20] L. Biasco, J. E. Massetti & M. Procesi - “An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS”, Comm. Math. Phys. 375 (2020) no. 3, p. 2089-2153 | DOI | MR | Zbl

[Bou93] J. Bourgain - “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations”, Geom. Funct. Anal. 3 (1993) no. 2, p. 107-156 | DOI | Zbl

[Bou96] J. Bourgain - “Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations”, Geom. Funct. Anal. 6 (1996) no. 2, p. 201-230 | DOI | Zbl

[Bou99] J. Bourgain - Global solutions of nonlinear Schrödinger equations, Amer. Math. Soc. Colloquium Publ., vol. 46, American Mathematical Society, Providence, RI, 1999 | DOI

[Bou00] J. Bourgain - “On diffusion in high-dimensional Hamiltonian systems and PDE”, J. Anal. Math. 80 (2000), p. 1-35 | DOI | MR | Zbl

[Caz03] T. Cazenave - Semilinear Schrödinger equations, Courant Lect. Notes in Math., vol. 10, American Mathematical Society, Providence, RI, 2003 | DOI

[CF12] R. Carles & E. Faou - “Energy cascades for NLS on the torus”, Discrete Contin. Dynam. Systems 32 (2012) no. 6, p. 2063-2077 | DOI | MR | Zbl

[CHL08a] D. Cohen, E. Hairer & C. Lubich - “Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations”, Numer. Math. 110 (2008) no. 2, p. 113-143 | DOI | MR | Zbl

[CHL08b] D. Cohen, E. Hairer & C. Lubich - “Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions”, Arch. Rational Mech. Anal. 187 (2008) no. 2, p. 341-368 | DOI | MR | Zbl

[CKS + 10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao - “Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation”, Invent. Math. 181 (2010) no. 1, p. 39-113 | DOI | Zbl

[Del09] J.-M. Delort - “On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus”, J. Anal. Math. 107 (2009), p. 161-194 | DOI | MR | Zbl

[Del12] J.-M. Delort - A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on 𝕊 1 , Astérisque, vol. 341, Société Mathématique de France, Paris, 2012 | Numdam

[Eli01] L. H. Eliasson - “Almost reducibility of linear quasi-periodic systems”, in Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, American Mathematical Society, Providence, RI, 2001, p. 679-705 | DOI | MR | Zbl

[FG13] E. Faou & B. Grébert - “A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus”, Anal. PDE 6 (2013) no. 6, p. 1243-1262 | DOI | Zbl

[FGL13] E. Faou, L. Gauckler & C. Lubich - “Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus”, Comm. Partial Differential Equations 38 (2013) no. 7, p. 1123-1140 | DOI | Zbl

[FI20] R. Feola & F. Iandoli - “A non-linear Egorov theorem and Poincaré-Birkhoff normal forms for quasi-linear pdes on the circle”, 2020 | arXiv

[FI21] R. Feola & F. Iandoli - “Long time existence for fully nonlinear NLS with small Cauchy data on the circle”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 22 (2021) no. 1, p. 109-182 | MR | Zbl

[GG12] P. Gérard & S. Grellier - “Effective integrable dynamics for a certain nonlinear wave equation”, Anal. PDE 5 (2012) no. 5, p. 1139-1155 | DOI | MR | Zbl

[GG17] P. Gérard & S. Grellier - The cubic Szegő equation and Hankel operators, Astérisque, vol. 389, Société Mathématique de France, Paris, 2017

[GGMP21] F. Giuliani, M. Guardia, P. Martin & S. Pasquali - “Chaotic-like transfers of energy in Hamiltonian PDEs”, Comm. Math. Phys. 384 (2021) no. 2, p. 1227-1290 | DOI | MR | Zbl

[GIP09] B. Grébert, R. Imekraz & É. Paturel - “Normal forms for semilinear quantum harmonic oscillators”, Comm. Math. Phys. 291 (2009) no. 3, p. 763-798 | DOI | MR | Zbl

[GK15] M. Guardia & V. Kaloshin - “Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation”, J. Eur. Math. Soc. (JEMS) 17 (2015) no. 1, p. 71-149, Erratum: Ibid. 19 (2017), no. 2, p. 601–602 | DOI

[Gré07] B. Grébert - “Birkhoff normal form and Hamiltonian PDEs”, in Partial differential equations and applications, Sémin. Congr., vol. 15, Société Mathématique de France, Paris, 2007, p. 1-46 | MR | Zbl

[GT12] B. Grébert & L. Thomann - “Resonant dynamics for the quintic nonlinear Schrödinger equation”, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012) no. 3, p. 455-477 | DOI | Numdam | Zbl

[Gua14] M. Guardia - “Growth of Sobolev norms in the cubic nonlinear Schrödinger equation with a convolution potential”, Comm. Math. Phys. 329 (2014) no. 1, p. 405-434 | DOI | MR | Zbl

[GVB11] B. Grébert & C. Villegas-Blas - “On the energy exchange between resonant modes in nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011) no. 1, p. 127-134 | DOI | Numdam | Zbl

[HP17] E. Haus & M. Procesi - “KAM for beating solutions of the quintic NLS”, Comm. Math. Phys. 354 (2017) no. 3, p. 1101-1132 | DOI | MR | Zbl

[HS17] M. Hofmanová & K. Schratz - “An exponential-type integrator for the KdV equation”, Numer. Math. 136 (2017) no. 4, p. 1117-1137 | DOI | MR | Zbl

[KP03] T. Kappeler & J. Pöschel - KdV & KAM, Ergeb. Math. Grenzgeb. (3), vol. 45, Springer-Verlag, Berlin, 2003 | DOI

[Mar86] V. A. Marchenko - Sturm-Liouville operators and applications, Operator Theory: Advances and Appl., vol. 22, Birkhäuser Verlag, Basel, 1986 | DOI | MR

[Neh77] N. N. Nehorošev - “An exponential estimate of the time of stability of nearly integrable Hamiltonian systems”, Russian Math. Surveys 32 (1977) no. 6, p. 1-65 | DOI

[ORS20] A. Ostermann, F. Rousset & K. Schratz - “Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces”, 2020, to appear in J. Eur. Math. Soc. (JEMS) | arXiv

[PT87] J. Pöschel & E. Trubowitz - Inverse spectral theory, Pure and Applied Math., vol. 130, Academic Press, Inc., Boston, MA, 1987

[Tri78] H. Triebel - Interpolation theory, function spaces, differential operators, North-Holland Math. Libr., vol. 18, Elsevier (North-Holland), Amsterdam, 1978

[Vla84] M. V. Vladimirov - “On the solvability of a mixed problem for a nonlinear equation of Schrödinger type”, Dokl. Akad. Nauk SSSR 275 (1984) no. 4, p. 780-783 | MR

[YZ14] X. Yuan & J. Zhang - “Long time stability of Hamiltonian partial differential equations”, SIAM J. Math. Anal. 46 (2014) no. 5, p. 3176-3222 | DOI | MR | Zbl

Cited by Sources: