Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$

Vedran Sohinger

Abstract

We consider Nonlinear Schrodinger 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. This iteration bound is different than the one used earlier in the work of Bourgain, and is less dependent on the structure of the nonlinearity. We first look at the defocusing NLS equation with nonlinearity of degree $\geq 5$. For the quintic NLS, Bourgain derives stronger bounds using different techniques. However, our approach works for higher nonlinearities, where the techniques of Bourgain don't seem to apply. Furthermore, we study variants of the defocusing cubic NLS in which the complete integrability is broken. Among this class of equations, we consider in particular the Hartree Equation, with sufficiently regular convolution potential. For most of the equations that come from modifying the defocusing cubic NLS, 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.

Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$

Abstract

We consider Nonlinear Schrodinger type equations on . 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. This iteration bound is different than the one used earlier in the work of Bourgain, and is less dependent on the structure of the nonlinearity. We first look at the defocusing NLS equation with nonlinearity of degree . For the quintic NLS, Bourgain derives stronger bounds using different techniques. However, our approach works for higher nonlinearities, where the techniques of Bourgain don't seem to apply. Furthermore, we study variants of the defocusing cubic NLS in which the complete integrability is broken. Among this class of equations, we consider in particular the Hartree Equation, with sufficiently regular convolution potential. For most of the equations that come from modifying the defocusing cubic NLS, 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.

Paper Structure

This paper contains 30 sections, 18 theorems, 562 equations.

Table of Contents

  1. Introduction.
  2. Setup of the Problem.
  3. Statement of the Main Results.
  4. Previously known results.
  5. Techniques of the Proof.
  6. Some notation.
  7. An observation about $X^{s,b}$ spaces.
  8. Quintic and Higher Order NLS.
  9. Definition of the $\mathcal{D}$ operator.
  10. A local-in-time estimate and an approximation lemma.
  11. Control on the increment of $\|\mathcal{D}u(t)\|_{L^2}^2$.
  12. Proof of Theorem \ref{['Theorem 1']} for $k \geq 3$.
  13. Remarks on the result of Bourgain.
  14. Modifications of the Cubic NLS.
  15. Modification 1: Hartree Equation.
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $k\geq 2$ be an integer and let $s\geq 1$ be a real number. Let $u$ be a global solution to (eq:NLS). Then, there exists a continuous function $C$, depending on $(s,k,E(\Phi),M(\Phi))$ such that, for all $t \in \mathbb{R}:$

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 29 more