Recent results on KAM for multidimensional PDEs
Journées équations aux dérivées partielles (2014), article no. 4, 12 p.

In this short overview I present some recent results about the KAM theory for multidimensional partial differential equations (PDEs) trying to avoid technicalities. In particular I will not state a precise KAM theorem but I will focus on the dynamical consequences for the PDEs: the existence and the stability (or not) of quasi periodic in time solutions. Concretely, I present the complete study of the nonlinear beam equation on the d-dimensional torus recently obtained in collaboration with H. Eliasson and S. Kuksin. When d2 we are able to construct explicit examples where the quasi periodic solutions are linearly unstable, a new feature in Hamiltonian PDEs that could complement recent results in weak turbulence theory.

DOI : https://doi.org/10.5802/jedp.107
Keywords: Multidimensional PDEs, Quasi periodic solutions, KAM theory, stable and unstable tori
@article{JEDP_2014____A4_0,
     author = {Gr\'ebert, Beno\^\i t},
     title = {Recent results on KAM for multidimensional PDEs},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.107},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2014____A4_0}
}
Grébert, Benoît. Recent results on KAM for multidimensional PDEs. Journées équations aux dérivées partielles (2014), article  no. 4, 12 p. doi : 10.5802/jedp.107. http://www.numdam.org/item/JEDP_2014____A4_0/

[1] V.I. Arnold, Mathematical methods in classical mechanics; 3d edition, Springer-Verlag, Berlin, 2006. | Zbl 0386.70001

[2] M. Berti, P. Bolle, Sobolev quasi periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity 25 (2012), 2579-2613. | MR 2967117 | Zbl 1262.35015

[3] M. Berti, P. Bolle, Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential, J. Eur. Math. Soc. 15 (2013), 229-286. | MR 2998835 | Zbl 1260.35196

[4] M. Berti, L. Corsi, M. Procesi, An Abstract Nash-Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds, Comm. Math. Phys (2014).

[5] M. Berti, M. Procesi, Nonlinear wave and Schrödinger equations on compact Lie groups and Homogeneous spaces, Duke Math. J. 159, 479-538 (2011). | MR 2831876 | Zbl 1260.37045

[6] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equation, Ann. Math. 148 (1998), 363-439. | MR 1668547 | Zbl 0928.35161

[7] J. Bourgain, Green’s function estimates for lattice Schrödinger operators and applications, Annals of Mathematical Studies, Princeton, 2004. | MR 2100420 | Zbl 1137.35001

[8] R. De La Llave, C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems, MPEJ 10 (2004), paper 5. | MR 2111299 | Zbl 1136.37349

[9] J.-M. Delort, Growth of Sobolev norms for solutions of time dependent Schrödinger operators with harmonic oscillator potential, Comm. Partial Differential Equations 39 (2014), 1Ð33. | MR 3169778 | Zbl 1291.35254

[10] L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, in Smooth ergodic theory and its applications (Seattle, WA, 1999), 679-705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001. | MR 1858550 | Zbl 1015.34028

[11] L. H. Eliasson, B. Grébert and S. B. Kuksin, KAM for the nonlinear beam equation 2: a normal form theorem, preprint. | MR 1115677

[12] L. H. Eliasson, B. Grébert and S. B. Kuksin, KAM for the nonlinear beam equation 1: small-amplitude solutions, preprint.

[13] L. H. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasiperiodic in time potentials, Comm. Math. Phys. 286 (2009), 125–135. | MR 2470926 | Zbl 1176.35141

[14] L. H. Eliasson and S. B. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. Math 172 (2010), 371-435. | MR 2680422 | Zbl 1201.35177

[15] E. Fermi, J. R. Pasta and S. M. Ulam, Studies of nonlinear problems. Collected works of E. Fermi, vol.2. Chicago University Press, Chicago, 1965.

[16] G. Gallavotti (editor). The Fermi-Pasta-Ulam problem Lectures Notes in Physics 728 Springer-Verlag, Berlin, 2008 | MR 2402016 | Zbl 1138.81004

[17] J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343–372. | MR 2200264 | Zbl 1103.37047

[18] J. Geng and J. You, KAM tori for higher dimensional beam equations with constant potentials, Nonlinearity, 19 (2006), 2405–2423. | MR 2260269 | Zbl 1190.37082

[19] B. Grébert, KAM for KG on 𝕊 2 and for the quantum harmonic oscillator on 2 , preprint arXiv:1410.8084.

[20] B. Grébert and L. Thomann, KAM for the Quantum Harmonic Oscillator, Comm. Math. Phys., 307 (2011), 383–427. | MR 2837120 | Zbl 1250.81033

[21] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21 (1987), 192–205. | MR 911772 | Zbl 0716.34083

[22] S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993. | MR 1290785 | Zbl 0784.58028

[23] S. B. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. Math. 143 (1996), 149–179. | MR 1370761 | Zbl 0847.35130

[24] T. Kappeler and J. Pöschel, KAM & KdV, Springer-Verlag, Berlin, 2003.

[25] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 119–148. | Numdam | MR 1401420 | Zbl 0870.34060

[26] C. Procesi and M. Procesi, A normal form of the nonlinear Schrödinger equation with analytic non–linearities, Comm. Math. Phys 312 (2012), 501-557. | MR 2917174 | Zbl 1277.35318

[27] C. Procesi and M. Procesi, A KAM algorithm for the resonant nonlinear Schrödinger equation, preprint 2013.

[28] W.-M. Wang, Nonlinear Schrödinger equations on the torus, Monograph, 156 pp, 2014 (submitted).