Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation. (English) Zbl 1311.35284
Summary: We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix \(s>1\). Recently J. E. Colliander et al. [Invent. Math. 181, No. 1, 39–113 (2010; Zbl 1197.35265)] proved the existence of solutions with \(s\)-Sobolev norm growing in time.
We establish the existence of solutions with polynomial time estimates. More exactly, there is \(c>0\) such that for any \(\mathcal K\gg 1\) we find a solution \(u\) and a time \(T\) such that \(\| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s}\). Moreover, the time \(T\) satisfies the polynomial bound \(0<T<\mathcal K^c\).
We establish the existence of solutions with polynomial time estimates. More exactly, there is \(c>0\) such that for any \(\mathcal K\gg 1\) we find a solution \(u\) and a time \(T\) such that \(\| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s}\). Moreover, the time \(T\) satisfies the polynomial bound \(0<T<\mathcal K^c\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K60 | Lattice dynamics; integrable lattice equations |
Keywords:
Hamiltonian partial differential equations; nonlinear Schrödinger equation; transfer of energy; growth of Sobolev norms; normal forms of Hamiltonian fixed pointsCitations:
Zbl 1197.35265References:
| [1] | Bambusi, D.: Long time stability of some small amplitude solutions in nonlinear Schrödinger equations. Comm. Math. Phys. 189, 205-226 (1997) · Zbl 0895.35095 · doi:10.1007/s002200050196 |
| [2] | Bambusi, D.: Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations. Math. Z. 230, 345-387 (1999) · Zbl 0928.35160 · doi:10.1007/PL00004696 |
| [3] | Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Comm. Math. Phys. 234, 253-285 (2003) · Zbl 1032.37051 · doi:10.1007/s00220-002-0774-4 |
| [4] | Bambusi, D., Grébert, B.: Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135, 507-567 (2006) · Zbl 1110.37057 · doi:10.1215/S0012-7094-06-13534-2 |
| [5] | Berti, M.: Nonlinear Oscillations of Hamiltonian PDEs. Progr. Nonlinear Differen- tial Equations Appl. 74, Birkhäuser Boston, Boston, MA (2007) · Zbl 1146.35002 |
| [6] | Berti, M., Biasco, L.: Branching of Cantor manifolds of elliptic tori and applications to PDEs. Comm. Math. Phys. 305, 741-796 (2011) · Zbl 1230.37092 · doi:10.1007/s00220-011-1264-3 |
| [7] | Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3, 107-156 (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020 |
| [8] | Bourgain, J.: On the growth in time of higher Sobolev norms of smooth solu- tions of Hamiltonian PDE. Int. Math. Res. Notices 1996, 277-304 · Zbl 0934.35166 · doi:10.1155/S1073792896000207 |
| [9] | Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. of Math. (2) 148, 363-439 (1998) · Zbl 0928.35161 · doi:10.2307/121001 |
| [10] | Bourgain, J.: On diffusion in high-dimensional Hamiltonian systems and PDE. J. Anal. Math. 80, 1-35 (2000) · Zbl 0964.35143 · doi:10.1007/BF02791532 |
| [11] | Bourgain, J.: Problems in Hamiltonian PDE’s. In: GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal., Special Volume, Part I, 32-56 (2000) · Zbl 1050.35016 |
| [12] | Bourgain, J.: Remarks on stability and diffusion in high-dimensional Hamiltonian sys- tems and partial differential equations. Ergodic Theory Dynam. Systems 24, 1331- 1357 (2004) · Zbl 1087.37056 · doi:10.1017/S0143385703000750 |
| [13] | Bronste\?ın, I. U., Kopanski\?ı, A. Ya.: Finitely smooth normal forms of vector fields in the vicinity of a rest point. In: Global Analysis-Studies and Applications, V, Lecture Notes in Math. 1520, Springer, Berlin, 157-172 (1992) |
| [14] | Bronste\?ın, I. U., Kopanski\?ı, A. Ya.: Smooth Invariant Manifolds and Normal Forms. World Sci., River Edge, NJ (1994) · Zbl 0974.34001 |
| [15] | Carles, R., Faou, E.: Energy cascade for NLS on d T . Discrete Contin. Dynam. Systems 32, 2063-2077 (2012) · Zbl 1238.35144 · doi:10.3934/dcds.2012.32.2063 |
| [16] | Catoire, F., Wang, W.-M.: Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Comm. Pure Appl. Anal. 9, 483-491 (2010) · Zbl 1189.35301 · doi:10.3934/cpaa.2010.9.483 |
| [17] | Colliander, J. E., Delort, J.-M., Kenig, C. E., Staffilani, G.: Bilinear estimates and appli- cations to 2D NLS. Trans. Amer. Math. Soc. 353, 3307-3325 (2001) · Zbl 0970.35142 · doi:10.1090/S0002-9947-01-02760-X |
| [18] | Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math. 181, 39-113 (2010) · Zbl 1197.35265 · doi:10.1007/s00222-010-0242-2 |
| [19] | Colliander, J., Kwon, S., Oh, T.: A remark on normal forms and the upside-down I- method for periodic NLS: growth of higher Sobolev norms. J. Anal. Math. 118, 55-82 (2012) · Zbl 1310.35213 · doi:10.1007/s11854-012-0029-z |
| [20] | Craig, W., Wayne, C. E.: Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. 46, 1409-1498 (1993) · Zbl 0794.35104 · doi:10.1002/cpa.3160461102 |
| [21] | Eliasson, L. H., Kuksin, S. B.: KAM for the nonlinear Schrödinger equation. Ann. of Math. (2) 172, 371-435 (2010) · Zbl 1201.35177 · doi:10.4007/annals.2010.172.371 |
| [22] | Fenichel, N.: Asymptotic stability with rate conditions. II. Indiana Univ. Math. J. 26, 81-93 (1977) · Zbl 0365.58012 · doi:10.1512/iumj.1977.26.26006 |
| [23] | Fenichel, N.: Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23, 1109-1137 (1973/74) · Zbl 0284.58008 · doi:10.1512/iumj.1974.23.23090 |
| [24] | Gérard, P., Grellier, S.: The cubic Szeg\Acute\Acute o equation. Ann. Sci. École Norm. Sup. (4) 43, 761-810 (2010) · Zbl 1228.35225 |
| [25] | Grébert, B., Imekraz, R., Paturel, E.: Normal forms for semilinear quantum harmonic oscillators. Comm. Math. Phys. 291, 763-798 (2009) · Zbl 1185.81073 · doi:10.1007/s00220-009-0800-x |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
