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Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$

Vedran Sohinger

Abstract

In this paper, we consider the cubic nonlinear Schrodinger equation, and the Hartree equation, with sufficiently regular convolution potential, both on the real line. We are interested in bounding the growth of high Sobolev norms of solutions to these equations. Since the cubic NLS is completely integrable, it makes sense to bound only the fractional Sobolev norms of solutions, whose initial data is of restricted smoothness. For the Hartree equation, we consider all Sobolev norms. For both equations, we derive our results by using an appropriate frequency decomposition. In the case of the cubic NLS, this method allows us to recover uniform bounds on the integral Sobolev norms, up to a factor of $t^{0+}$. For the Hartree equation, we use the same method as in our previous work on $S^1$, and the improved Strichartz estimate to obtain a better bound than we previously obtained in the periodic setting.

Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$

Abstract

In this paper, we consider the cubic nonlinear Schrodinger equation, and the Hartree equation, with sufficiently regular convolution potential, both on the real line. We are interested in bounding the growth of high Sobolev norms of solutions to these equations. Since the cubic NLS is completely integrable, it makes sense to bound only the fractional Sobolev norms of solutions, whose initial data is of restricted smoothness. For the Hartree equation, we consider all Sobolev norms. For both equations, we derive our results by using an appropriate frequency decomposition. In the case of the cubic NLS, this method allows us to recover uniform bounds on the integral Sobolev norms, up to a factor of . For the Hartree equation, we use the same method as in our previous work on , and the improved Strichartz estimate to obtain a better bound than we previously obtained in the periodic setting.

Paper Structure

This paper contains 16 sections, 15 theorems, 309 equations.

Table of Contents

  1. Introduction.
  2. Statement of the problem and of the main results:
  3. Motivation for the problem and previously known results:
  4. Main ideas of the proofs:
  5. Main ideas of the proof of Theorem \ref{['Theorem 1']}:
  6. Main ideas of the proof of Theorem \ref{['Theorem 2']}:
  7. Notation and known facts.
  8. The cubic nonlinear Schrödinger equation.
  9. Basic facts about the equation:
  10. An Iteration bound and Proof of Theorem \ref{['Theorem 1']}:
  11. The Hartree equation.
  12. Basic facts about the equation and definition of the $\mathcal{D}$ operator.
  13. An Iteration bound and proof of Theorem \ref{['Theorem 2']}
  14. Appendix A: Auxiliary results for the cubic nonlinear Schrödinger equation.
  15. Appendix B: Auxiliary results for the Hartree equation.
  16. ...and 1 more sections

Key Result

Theorem 1.1

(Bound for the Cubic NLS) Suppose $s>1$ is not an integer. Let $\alpha:=s-\lfloor s \rfloor$ denote the fractional part of $s$. Suppose $\Phi \in H^s(\mathbb{R})$, and let $u$ denote the global solution to the corresponding problem (eq:cubicnls). Then, there exists a continuous function $F_s: H^s \r

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 17 more