Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $S^1$

Abstract

We consider nonlinear Schrödinger-type equations on $S^1$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of their solutions. The key is to derive an iteration bound based on a frequency decomposition of the solution, which is different than the iteration bound first used by Bourgain in [4]. We first look at the NLS equation with nonlinearity of degree $\geq 5$. For $q=5$, Bourgain in [9] derives stronger bounds using different techniques. However, our approach works for higher nonlinearities, where the techniques from [9] don't seem to apply. Furthermore, we study non-integrable modifications of the cubic NLS, among which is the Hartree equation, with sufficiently regular convolution potential. For most of the equations obtained this way, we obtain better bounds than for the other equations, due to the fact that we can use higher modified energies, as in the work of the I-Team [18, 20].