Finite dimensional invariant KAM tori for tame vector fields
HTML articles powered by AMS MathViewer
- by Livia Corsi, Roberto Feola and Michela Procesi PDF
- Trans. Amer. Math. Soc. 372 (2019), 1913-1983 Request permission
Abstract:
We discuss a Nash-Moser/KAM algorithm for the construction of invariant tori for tame vector fields. Similar algorithms have been studied widely both in finite and infinite dimensional contexts: we are particularly interested in the second case where tameness properties of the vector fields become very important. We focus on the formal aspects of the algorithm and particularly on the minimal hypotheses needed for convergence. We discuss various applications where we show how our algorithm allows one to reduce to solving only linear forced equations. We remark that our algorithm works at the same time in analytic and Sobolev classes.References
- V. I. Arnol′d, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192 (Russian). MR 0170705
- Pietro Baldi, Periodic solutions of forced Kirchhoff equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 1, 117–141. MR 2512203
- Pietro Baldi, Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 1, 33–77 (English, with English and French summaries). MR 3011291, DOI 10.1016/j.anihpc.2012.06.001
- Pietro Baldi, Massimiliano Berti, and Riccardo Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann. 359 (2014), no. 1-2, 471–536. MR 3201904, DOI 10.1007/s00208-013-1001-7
- Pietro Baldi, Massimiliano Berti, and Riccardo Montalto, KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 6, 1589–1638. MR 3569244, DOI 10.1016/j.anihpc.2015.07.003
- Massimiliano Berti and Luca Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys. 305 (2011), no. 3, 741–796. MR 2819413, DOI 10.1007/s00220-011-1264-3
- Massimiliano Berti, Luca Biasco, and Michela Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 2, 301–373 (2013) (English, with English and French summaries). MR 3112201, DOI 10.24033/asens.2190
- Massimiliano Berti, Luca Biasco, and Michela Procesi, KAM for reversible derivative wave equations, Arch. Ration. Mech. Anal. 212 (2014), no. 3, 905–955. MR 3187681, DOI 10.1007/s00205-014-0726-0
- M. Berti and P. Bolle, Quasi-periodic solutions for autonomous NLW on $\mathbb T^d$ with a multiplicative potential. In preparation.
- Massimiliano Berti and Philippe Bolle, Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity 25 (2012), no. 9, 2579–2613. MR 2967117, DOI 10.1088/0951-7715/25/9/2579
- Massimiliano Berti and Philippe Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on $\Bbb T^d$ with a multiplicative potential, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 229–286. MR 2998835, DOI 10.4171/JEMS/361
- Massimiliano Berti, Livia Corsi, and Michela Procesi, An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds, Comm. Math. Phys. 334 (2015), no. 3, 1413–1454. MR 3312439, DOI 10.1007/s00220-014-2128-4
- M. Berti and R. Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, preprint, arXiv:1602.02411, 2016. To appear in Mem. Amer. Math. Soc.
- Massimiliano Berti and Philippe Bolle, A Nash-Moser approach to KAM theory, Hamiltonian partial differential equations and applications, Fields Inst. Commun., vol. 75, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, pp. 255–284. MR 3445505, DOI 10.1007/978-1-4939-2950-4_{9}
- Jean Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices 11 (1994), 475ff., approx. 21 pp.}, issn=1073-7928, review= MR 1316975, doi=10.1155/S1073792894000516, DOI 10.1155/S1073792894000516
- J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math. (2) 148 (1998), no. 2, 363–439. MR 1668547, DOI 10.2307/121001
- J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575, DOI 10.1090/coll/046
- Jean Bourgain, Periodic solutions of nonlinear wave equations, Harmonic analysis and partial differential equations (Chicago, IL, 1996) Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 69–97. MR 1743856
- J. Bourgain, Green’s function estimates for lattice Schrödinger operators and applications, Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton, NJ, 2005. MR 2100420, DOI 10.1515/9781400837144
- Luigi Chierchia and Jiangong You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000), no. 2, 497–525. MR 1754527, DOI 10.1007/s002200050824
- Livia Corsi, Emanuele Haus, and Michela Procesi, A KAM result on compact Lie groups, Acta Appl. Math. 137 (2015), 41–59. MR 3343372, DOI 10.1007/s10440-014-9990-0
- Walter Craig and C. Eugene Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), no. 11, 1409–1498. MR 1239318, DOI 10.1002/cpa.3160461102
- L. Hakan Eliasson and Sergei B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math. (2) 172 (2010), no. 1, 371–435. MR 2680422, DOI 10.4007/annals.2010.172.371
- J. Fejoz, Periodic and quasi-periodic motions in the many-body problem, mémoire d’habilitation à diriger des recherches, 2010.
- R. Feola, Quasi-periodic solutions for fully nonlinear NLS, PhD thesis, University “La Sapienza”, 2016.
- R. Feola and M. Procesi, KAM for quasi-linear autonomous NLS, preprint, arXiv:1705.07287, 2017.
- Roberto Feola and Michela Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations 259 (2015), no. 7, 3389–3447. MR 3360677, DOI 10.1016/j.jde.2015.04.025
- Jiansheng Geng, Xindong Xu, and Jiangong You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math. 226 (2011), no. 6, 5361–5402. MR 2775905, DOI 10.1016/j.aim.2011.01.013
- Jiansheng Geng and Jiangong You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys. 262 (2006), no. 2, 343–372. MR 2200264, DOI 10.1007/s00220-005-1497-0
- Filippo Giuliani, Quasi-periodic solutions for quasi-linear generalized KdV equations, J. Differential Equations 262 (2017), no. 10, 5052–5132. MR 3612536, DOI 10.1016/j.jde.2017.01.021
- Benoît Grébert and Eric Paturel, KAM for the Klein Gordon equation on $\Bbb {S}^d$, Boll. Unione Mat. Ital. 9 (2016), no. 2, 237–288. MR 3502160, DOI 10.1007/s40574-016-0072-2
- G. Iooss, P. I. Plotnikov, and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal. 177 (2005), no. 3, 367–478. MR 2187619, DOI 10.1007/s00205-005-0381-6
- Thomas Kappeler and Riccardo Montalto, Canonical coordinates with tame estimates for the defocusing NLS equation on the circle, Int. Math. Res. Not. IMRN 5 (2018), 1473–1531. MR 3801469, DOI 10.1093/imrn/rnw233
- Thomas Kappeler and Jürgen Pöschel, KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 45, Springer-Verlag, Berlin, 2003. MR 1997070, DOI 10.1007/978-3-662-08054-2
- A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 527–530 (Russian). MR 0068687
- S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 22–37, 95 (Russian). MR 911772
- Sergei B. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys. 10 (1998), no. 3, ii+64. MR 1754991
- Sergej Kuksin and Jürgen Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2) 143 (1996), no. 1, 149–179. MR 1370761, DOI 10.2307/2118656
- Jianjun Liu and Xiaoping Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys. 307 (2011), no. 3, 629–673. MR 2842962, DOI 10.1007/s00220-011-1353-3
- Riccardo Montalto, Quasi-periodic solutions of forced Kirchhoff equation, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 1, Paper No. 9, 71. MR 3603787, DOI 10.1007/s00030-017-0432-3
- Jürgen Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 265–315. MR 199523
- Jürgen Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967), 136–176. MR 208078, DOI 10.1007/BF01399536
- Jürgen Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 119–148. MR 1401420
- Jürgen Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv. 71 (1996), no. 2, 269–296. MR 1396676, DOI 10.1007/BF02566420
- Jürgen Pöschel, A lecture on the classical KAM theorem, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 707–732. MR 1858551, DOI 10.1090/pspum/069/1858551
- C. Procesi and M. Procesi, A KAM algorithm for the resonant non-linear Schrödinger equation, Adv. Math. 272 (2015), 399–470. MR 3303238, DOI 10.1016/j.aim.2014.12.004
- M. Procesi and C. Procesi, Reducible quasi-periodic solutions for the non linear Schrödinger equation, Boll. Unione Mat. Ital. 9 (2016), no. 2, 189–236. MR 3502159, DOI 10.1007/s40574-016-0066-0
- Helmut Rüssmann, Nondegeneracy in the perturbation theory of integrable dynamical systems, Number theory and dynamical systems (York, 1987) London Math. Soc. Lecture Note Ser., vol. 134, Cambridge Univ. Press, Cambridge, 1989, pp. 5–18. MR 1043702, DOI 10.1017/CBO9780511661983.002
- Mikhail B. Sevryuk, The reversible context 2 in KAM theory: the first steps, Regul. Chaotic Dyn. 16 (2011), no. 1-2, 24–38. MR 2774376, DOI 10.1134/S1560354710520035
- C. Eugene Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), no. 3, 479–528. MR 1040892
- E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1976), no. 1, 49–111. MR 426055, DOI 10.1002/cpa.3160290104
- Jing Zhang, Meina Gao, and Xiaoping Yuan, KAM tori for reversible partial differential equations, Nonlinearity 24 (2011), no. 4, 1189–1228. MR 2776117, DOI 10.1088/0951-7715/24/4/010
Additional Information
- Livia Corsi
- Affiliation: Department of Mathematics, Emory University, Atlanta, Georgia 30307
- MR Author ID: 857630
- Email: lcorsi@emory.edu
- Roberto Feola
- Affiliation: Dipartimento di Matematica, SISSA-Trieste, 34136 Trieste, Italy
- Address at time of publication: Laboratoire de Mathématiques J. Leray, Université de Nantes, Nantes, France
- MR Author ID: 1011573
- Email: roberto.feola@univ-nantes.fr
- Michela Procesi
- Affiliation: Dipartimento di Matematica e Fisica, Università di Roma Tre, 00146 Roma RM, Italy
- MR Author ID: 681901
- Email: procesi@mat.uniroma3.it
- Received by editor(s): February 10, 2017
- Received by editor(s) in revised form: June 13, 2018
- Published electronically: May 9, 2019
- Additional Notes: This research was supported by the European Research Council under FP7 “Hamiltonian PDEs and small divisor problems: a dynamical systems approach”, by PRIN2012 “Variational and perturbative aspects of non linear differential problems”, by the NSF grant DMS-1500943, and by McMaster University.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1913-1983
- MSC (2010): Primary 37K55, 37J40
- DOI: https://doi.org/10.1090/tran/7699
- MathSciNet review: 3976581