Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

Vedran Sohinger

doi:10.3934/dcds.2012.32.3733

Article Contents

2012, Volume 32, Issue 10: 3733-3771. Doi: 10.3934/dcds.2012.32.3733

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Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations

Vedran Sohinger^1,

1.
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

Received: April 30, 2010

Revised: December 31, 2011

Published: September 2012

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Abstract

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we had previously used in [49, 50] to study the analogous problem in one dimension. Since we are working in two dimensions, a more detailed analysis of the resonant frequencies is needed, as was previously used in the work of Colliander-Keel-Staffilani-Takaoka-Tao [19].

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Mathematics Subject Classification: 35Q55.

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