A remark on normal forms and the "upside-down" I-method for periodic NLS: Growth of higher Sobolev norms

James Colliander, Soonsik Kwon, Tadahiro Oh

Research output: Contribution to journalArticlepeer-review

Abstract

We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s} \lesssim (1+|t|)^{\alpha (s-1)+} with \alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with \alpha = 1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, we concretely compute the terms arising in the first few steps of the normal form reduction and prove the above estimate with \alpha = 4/9. These results improve the previously known results (except for the quintic case.) In Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers. with alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with alpha = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with alpha = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.

Original languageEnglish
Pages (from-to)55-82
Number of pages28
JournalJournal d'Analyse Mathématique
Volume118
DOIs
Publication statusPublished - Oct 2012

Keywords

  • Schrodinger equation
  • normal form
  • upside-down I-method
  • growth of Sobolev norm

Cite this