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Polynomial growth of Sobolev norms of solutions of the fractional NLS equation on \T^d

Jiajun Wang

Abstract

In this paper, we prove polynomial growth bounds for the Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on the torus \T^d (d \ge 2), following and extending a result of Joseph Thirouin on \T [Thi17]. The key ingredient is the establishment of Strichartz estimates for the fractional Schrödinger equation on \T^d. To this end, we employ uniform estimates for oscillatory integrals to overcome the lack of uniformity that arises in higher dimensions.

Polynomial growth of Sobolev norms of solutions of the fractional NLS equation on \T^d

Abstract

In this paper, we prove polynomial growth bounds for the Sobolev norms of solutions to the fractional nonlinear Schrödinger equation on the torus \T^d (d \ge 2), following and extending a result of Joseph Thirouin on \T [Thi17]. The key ingredient is the establishment of Strichartz estimates for the fractional Schrödinger equation on \T^d. To this end, we employ uniform estimates for oscillatory integrals to overcome the lack of uniformity that arises in higher dimensions.

Paper Structure

This paper contains 5 sections, 22 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Strichartz estimates for the fractional Schrödinger equation
  3. Bourgain spaces and multi-linear estimates
  4. Well-poseness theory of fractional nonlinear Schrödinger equation
  5. Polynomial growth of Sobolev norms

Key Result

Theorem 1.1

For $1\le p\le \infty$, if $\alpha>1$, $\gamma>\frac{\gamma_{\alpha,p}}{2}$, we have the Strichartz estimate Similarly, if $0<\alpha<1$, $\gamma>\frac{\ell_{\alpha,p}}{2}$, the above estimate also holds.

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 46 more