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

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.