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Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations

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dc.contributor.advisor Gigliola Staffilani. en_US Sohinger, Vedran en_US
dc.contributor.other Massachusetts Institute of Technology. Dept. of Mathematics. en_US 2011-12-19T19:01:05Z 2011-12-19T19:01:05Z 2011 en_US 2011 en_US
dc.description Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011. en_US
dc.description Cataloged from PDF version of thesis. en_US
dc.description Includes bibliographical references (p. 265-273). en_US
dc.description.abstract In this thesis, we study the growth of Sobolev norms of global solutions of solutions to nonlinear Schrödinger type equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate on the low-to high frequency cascade which can occur due to the nonlinear evolution. In our work, we present two possible frequency decomposition methods which allow us to obtain polynomial bounds on the high Sobolev norms of the solutions to the equations we are considering. The first method is a high regularity version of the I-method previously used by Colliander, Keel, Staffilani, Takaoka, and Tao and it allows us to treat a wide range of equations, including the power type NLS equation and the Hartree equation with sufficiently regular convolution potential, as well as the Gross-Pitaevskii equation for dipolar quantum gases in the physically relevant 3D setting. The other method is based on a rough cut-off in frequency and it allows us to bound the growth of fractional Sobolev norms of the completely integrable defocusing cubic NLS on the real line. en_US
dc.description.statementofresponsibility by Vedran Sohinger. en_US
dc.format.extent 273 p. en_US
dc.language.iso eng en_US
dc.publisher Massachusetts Institute of Technology en_US
dc.rights M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. en_US
dc.rights.uri en_US
dc.subject Mathematics. en_US
dc.title Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations en_US
dc.type Thesis en_US Ph.D. en_US
dc.contributor.department Massachusetts Institute of Technology. Dept. of Mathematics. en_US
dc.identifier.oclc 767996445 en_US

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