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

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We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish parallel to u(t)parallel to H-s less than or similar to(1+vertical bar t vertical bar)(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 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.
Publisher
SPRINGER
Issue Date
2012-10
Language
English
Article Type
Article
Keywords

NONLINEAR SCHRODINGER-EQUATION; BOUNDS

Citation

JOURNAL D ANALYSE MATHEMATIQUE, v.118, pp.55 - 82

ISSN
0021-7670
DOI
10.1007/s11854-012-0029-z
URI
http://hdl.handle.net/10203/104407
Appears in Collection
MA-Journal Papers(저널논문)
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