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Symmetric hyperbolic Schrödinger equations on tori

Baoping Liu, Xu Zheng

Abstract

In this paper, we study the symmetric hyperbolic Schrödinger equations in the periodic setting. First, we prove full range Strichartz estimates on general tori by adapting Bourgain's major arc method. The result is sharp for rational tori. Second, on two-dimensional rational tori, we establish optimal local well-posedness for two hyperbolic nonlinear Schrödinger (HNLS) equations: the septic HNLS and the hyperbolic-elliptic Davey-Stewartson system.

Symmetric hyperbolic Schrödinger equations on tori

Abstract

In this paper, we study the symmetric hyperbolic Schrödinger equations in the periodic setting. First, we prove full range Strichartz estimates on general tori by adapting Bourgain's major arc method. The result is sharp for rational tori. Second, on two-dimensional rational tori, we establish optimal local well-posedness for two hyperbolic nonlinear Schrödinger (HNLS) equations: the septic HNLS and the hyperbolic-elliptic Davey-Stewartson system.

Paper Structure

This paper contains 13 sections, 22 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Strichartz Estimates
  3. Notations
  4. Strichartz Estimate
  5. Bilinear Estimates
  6. Line concentrated functions
  7. Off-diagonal estimates
  8. Local Well-posedness
  9. Function spaces
  10. Bilinear and trilinear estimate
  11. Estimate for the Duhamel term
  12. Contractions
  13. Ill-posedness

Key Result

Theorem 1.1

For $v=\frac{d}{2}$ and any $\phi\in {L^2_{x}(\mathbb T^{d})}$, we have that $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Relation between $\beta_{d,v}(p)$ and $p$

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: bour1989israel
  • ...and 39 more