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On sharp Strichartz estimate for hyperbolic Schrödinger equation on $\mathbb{T}^3$

Baoping Liu, Xu Zheng

TL;DR

This work establishes a sharp Strichartz estimate for the hyperbolic Schrödinger equation on the 3-torus, removing the $N^{\varepsilon}$ loss that appears in prior decoupling-based results. The authors reduce the endpoint to an $L^4_{t,x}$ bound and tackle it via incidence geometry, analyzing frequency parallelograms constrained by a cone and exploiting Szemerédi–Trotter type bounds. Building on this, they derive optimal local well-posedness results for nonlinear hyperbolic Schrödinger equations on $\mathbb T^3$ in the natural Sobolev scales, and demonstrate ill-posedness at the critical index for the cubic case. The approach blends harmonic analysis on compact manifolds with incidence-geometry techniques, offering precise frequency-localized control and potential applicability to related dispersive PDEs on tori.

Abstract

We prove the sharp Strichartz estimate for hyperbolic Schrödinger equation on $\mathbb{T}^3 $ via an incidence geometry approach. As application, we obtain optimal local well-posedness of nonlinear hyperbolic Schrödinger equations.

On sharp Strichartz estimate for hyperbolic Schrödinger equation on $\mathbb{T}^3$

TL;DR

This work establishes a sharp Strichartz estimate for the hyperbolic Schrödinger equation on the 3-torus, removing the loss that appears in prior decoupling-based results. The authors reduce the endpoint to an bound and tackle it via incidence geometry, analyzing frequency parallelograms constrained by a cone and exploiting Szemerédi–Trotter type bounds. Building on this, they derive optimal local well-posedness results for nonlinear hyperbolic Schrödinger equations on in the natural Sobolev scales, and demonstrate ill-posedness at the critical index for the cubic case. The approach blends harmonic analysis on compact manifolds with incidence-geometry techniques, offering precise frequency-localized control and potential applicability to related dispersive PDEs on tori.

Abstract

We prove the sharp Strichartz estimate for hyperbolic Schrödinger equation on via an incidence geometry approach. As application, we obtain optimal local well-posedness of nonlinear hyperbolic Schrödinger equations.

Paper Structure

This paper contains 14 sections, 17 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Strichartz Estimate
  3. Notations
  4. Facts from incidence geometry
  5. Preparation
  6. The contributions of parallelograms with side on the cone
  7. The contributions of parallelograms without side on the cone
  8. Proof of Theorem \ref{['sharp T3 L4 hyperbolic Strichartz']}
  9. Sharpness of Strichartz estimate
  10. Local Well-posedness
  11. Function spaces
  12. Multilinear estimates
  13. Proof of Theorem \ref{['local well-posedness of HNLS on T^3']}
  14. Proof of Theorem \ref{['ill-posedness of HNLS on T^3']}

Key Result

Theorem 1.1

For $\phi \in L^2(\mathbb T^3)$, we have that

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Szemeredi-TrotterPach-Sharir2004Points-lines incidences
  • Corollary 2.2: Pach-Sharir2004
  • Theorem 2.3: Points-circles incidences on sphere
  • proof
  • Remark 2.4
  • ...and 30 more