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March  2010, 9(2): 483-491. doi: 10.3934/cpaa.2010.9.483

Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori

F. Catoire 1, and  W. M. Wang 1,

1. 

Département de Mathématique, Université Paris Sud, 91405 Orsay Cedex, France, France

Received  March 2009 Revised  August 2009 Published  December 2009

We prove Strichartz estimates on general flat $d$-torus for arbitrary $d$. Using these estimates, we prove local wellposedness for the cubic nonlinear Schrödinger equations in appropriate Sobolev spaces. In dimensions $2$ and $3$, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions.
Keywords: Sobolev norms., Nonlinear Schrödinger equations, Strichartz estimates.
Mathematics Subject Classification: 35xx, 42x.
Citation: F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure and Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483
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