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Growth of Sobolev Norms in the Cubic Nonlinear Schrödinger Equation with a Convolution Potential

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Abstract

Fix s > 1. Colliander et al. proved in (Invent Math 181:39–113, 2010) the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus whose s-Sobolev norm undergoes arbitrarily large growth as time evolves. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in Guardia and Kaloshin (Growth of Sobolev norms in the cubic defocusing Nonlinear Schrödinger Equation. To appear in the Journal of the European Mathematical Society, 2012).

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References

  1. Bambusi, D., Grébert, B.:Forme normale pour NLS en dimension quelconque. C. R. Math. Acad. Sci. Paris. 337(6), 409–414 (2003)

  2. Bambusi D., Grébert B.: Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135(3), 507–567 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)

    Google Scholar 

  4. Bourgain, J.: On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Intern. Math. Res. Notices 6, 277–304 (1996)

    Article  MathSciNet  Google Scholar 

  5. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. (2) 148(2), 363–439 (1998)

    Google Scholar 

  6. Bourgain, J.: Problems in Hamiltonian PDE’s. Geom. Funct. Anal., Special Volume, Part I, pp. 32–56 (2000). GAFA 2000 (Tel Aviv, 1999)

  7. Bourgain, J.: Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations. Ergodic Theory Dynam. Syst. 24(5), 1331–1357 (2004)

    Google Scholar 

  8. Colliander, J.E., Delort, J.-M., Kenig, C.E., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. 353(8), 3307–3325 (electronic), (2001)

    Google Scholar 

  9. Carles, R., Faou, E.: Energy cascades for NLS on the torus. Discret. Contin. Dyn. Syst. 32(6), 2063–2077 (2012)

    Google Scholar 

  10. 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(1), 55–82 (2012)

    Google Scholar 

  11. 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(1), 39–113 (2010)

    Google Scholar 

  12. Catoire, F., Wang, W.M.: Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Commun. Pure Appl. Anal. 9(2), 483–491 (2010)

    Google Scholar 

  13. Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. (2) 172(1), 371–435 (2010)

    Google Scholar 

  14. Ebin D.G., Marsden J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. (2) 92, 102–163 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  15. Faou, E., Grebert, B.: A Nekhoroshev type theorem for the nonlinear Schrdinger equation on the d-dimensional torus. To appear in Analysis and PDE. Preprint available at http://arxiv.org/abs/1003.4845 (2010)

  16. Gérard, P., Grellier, S.: The cubic Szegő equation. Ann. Sci. Éc. Norm. Supér. (4) 43(5), 761–810 (2010)

    Google Scholar 

  17. Gérard P., Grellier S.: Effective integrable dynamics for a certain nonlinear wave equation. Anal. PDE 5(5), 1139–1155 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guardia, M., Kaloshin, V.:Growth of Sobolev norms in the cubic defocusing Nonlinear Schrödinger Equation. To appear in the Journal of the European Mathematical Society. Preprint available at http://arxiv.org/abs/1205.5188 (2012)

  19. Gauckler, L., Lubich, C.:Nonlinear Schrödinger equations and their spectral semi-discretizations over long times. Found. Comput. Math. 10(2), 141–169 (2010)

    Google Scholar 

  20. Hani, Z.: Global and dynamical aspects of nonlinear Schrödinger equations on compact manifolds. Ph.D. thesis UCLA (2011)

  21. Hani, Z.: Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrdinger equations. To appear in Archives for Rational Mechanics and Analysis (ARMA). Preprint available at http://arxiv.org/abs/1210.7509 (2012)

  22. Kuksin, S.B.: Oscillations in space-periodic nonlinear Schrödinger equations. Geom. Funct. Anal. 7(2), 338–363 (1997)

    Google Scholar 

  23. Pocovnicu, O.: Explicit formula for the solution of the Szegö equation on the real line and applications. Discret. Contin. Dyn. Syst. 31(3), 607–649 (2011)

    Google Scholar 

  24. Pocovnicu, O.: First and second order approximations for a nonlinear wave equation. J. Dynam. Differ. Equ. 25(2), 305–333 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sohinger, V.: Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on S 1. Differ. Integr. Equ. 24(7–8), 653–718 (2011)

  26. Sohinger, V.: Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on \({\mathbb{R}}\). Indiana Univ. Math. J. 60(5), 1487–1516 (2011)

    Google Scholar 

  27. Sohinger, V.: Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discret. Contin. Dyn. Syst. 32(10), 3733–3771 (2012)

    Google Scholar 

  28. Staffilani, G.: On the growth of high sobolev norms of solutions for kdv and Schrödinger equations. Duke Math. J. 86(1), 109–142 (1997)

    Google Scholar 

  29. Staffilani, G.: Quadratic forms for a 2-D semilinear Schrödinger equation. Duke Math. J. 86(1), 79–107 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhong, S.: The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds. J. Differ. Equ. 245(2), 359–376 (2008)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université Paris 7 Denis Diderot, 75205, Paris, France

    Marcel Guardia

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  1. Marcel Guardia
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Correspondence to Marcel Guardia.

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Communicated by W. Schlag

The author is partially supported by the Spanish MCyT/FEDER grants MTM2009-06973 and MTM2012-31714 and the Catalan SGR grant 2009SGR859.

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Guardia, M. Growth of Sobolev Norms in the Cubic Nonlinear Schrödinger Equation with a Convolution Potential. Commun. Math. Phys. 329, 405–434 (2014). https://doi.org/10.1007/s00220-014-1977-1

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