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Local well-posedness for nonlinear Schrödinger equations on compact product manifolds

Yunfeng Zhang

TL;DR

This paper develops a unified framework for local well-posedness of nonlinear Schrödinger equations on compact product manifolds built from spheres and tori by establishing multilinear Strichartz estimates via joint spectral projector bounds. Central to the approach are the multi-linear joint spectral projector estimates (Theorem mljspe) that respect the product structure and are sharper than full Laplacian bounds when multiple sphere factors are present, and the resulting multilinear Strichartz bounds (Theorem mlS) that feed into local well-posedness results (Theorem lwp) across subcritical, critical, and near-critical regimes. A key technical input is the sharp exponential-sum bound (Lemma exp) on torus factors, enabling control over time-restricted linear Schrödinger evolution. The work extends the Burq--Gérard--Tzvetkov program to product geometries, includes an analysis for the cubic NLS at critical regularity on specific products, and raises several open questions about sharpness and extensions to Euclidean factors. Overall, the results advance understanding of NLS dynamics on composite geometries and provide a robust toolkit for handling critical and subcritical regimes on product manifolds.

Abstract

We prove new local well-posedness results for nonlinear Schrödinger equations posed on a general product of spheres and tori, by the standard approach of multi-linear Strichartz estimates. To prove these estimates, we establish and utilize multi-linear bounds for the joint spectral projector associated to the Laplace--Beltrami operators on the individual sphere factors of the product manifold. To treat the particular case of the cubic NLS on a product of two spheres at critical regularity, we prove a sharp $L^\infty_xL^p_t$ estimate of the solution to the linear Schrödinger equation on the two-torus.

Local well-posedness for nonlinear Schrödinger equations on compact product manifolds

TL;DR

This paper develops a unified framework for local well-posedness of nonlinear Schrödinger equations on compact product manifolds built from spheres and tori by establishing multilinear Strichartz estimates via joint spectral projector bounds. Central to the approach are the multi-linear joint spectral projector estimates (Theorem mljspe) that respect the product structure and are sharper than full Laplacian bounds when multiple sphere factors are present, and the resulting multilinear Strichartz bounds (Theorem mlS) that feed into local well-posedness results (Theorem lwp) across subcritical, critical, and near-critical regimes. A key technical input is the sharp exponential-sum bound (Lemma exp) on torus factors, enabling control over time-restricted linear Schrödinger evolution. The work extends the Burq--Gérard--Tzvetkov program to product geometries, includes an analysis for the cubic NLS at critical regularity on specific products, and raises several open questions about sharpness and extensions to Euclidean factors. Overall, the results advance understanding of NLS dynamics on composite geometries and provide a robust toolkit for handling critical and subcritical regimes on product manifolds.

Abstract

We prove new local well-posedness results for nonlinear Schrödinger equations posed on a general product of spheres and tori, by the standard approach of multi-linear Strichartz estimates. To prove these estimates, we establish and utilize multi-linear bounds for the joint spectral projector associated to the Laplace--Beltrami operators on the individual sphere factors of the product manifold. To treat the particular case of the cubic NLS on a product of two spheres at critical regularity, we prove a sharp estimate of the solution to the linear Schrödinger equation on the two-torus.

Paper Structure

This paper contains 8 sections, 15 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mljspe']}
  3. Review of Burq--Gérard--Tzvetkov BGT052
  4. Multi-linear estimates
  5. Proof of Theorem \ref{['mls']}
  6. Proof of Theorem \ref{['lwp']}
  7. Proof of Lemma \ref{['exp']}
  8. Some open questions

Key Result

Theorem 1.1

Let $M$ be a product of spheres and tori: $M=\mathbb{S}^{d_1}\times\mathbb{S}^{d_2}\times\cdots\times \mathbb{S}^{d_{r_0}}\times \mathbb{T}^{r_1}$, with $d_i\geq 2$ ($i=1,2,\ldots,r_0$) and $r:=r_0+r_1\geq 2$. Let $r_2$ (respectively, $r_3$) be the number of 2-sphere (respectively, 3-sphere) factors (ii) For $k=1$, $r=2$, we have for all $\varepsilon>0$ (iii) For $k=1$ and the special case $M=\mat

Theorems & Definitions (23)

  • Theorem 1.1: Multi-linear Strichartz estimate
  • Theorem 1.2: Main result
  • Remark 1.3
  • Theorem 1.4: Multi-linear spectral projector estimate of Burq--Gérard--Tzvetkov
  • Theorem 1.5: Multi-linear joint spectral projector estimate
  • Conjecture 1.6
  • Lemma 1.7
  • Lemma 2.1: Lemma 2.3 of BGT052
  • Lemma 2.2: Lemma 2.10 of BGT052
  • Lemma 2.3: Lemma 2.11 of BGT052
  • ...and 13 more