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Refined Strichartz estimates and their orthornomal counterparts for Schrödinger equations on torus

Divyang G. Bhimani, Subhash. R. Choudhary, S. S. Mondal

TL;DR

This work develops refined Strichartz estimates and their orthonormal counterparts for the Schrödinger equation on the torus under a partial regularity framework, proving improved mixed Lebesgue regularity for initial data in the spaces $X_k^s$ and extending to infinite systems of orthonormal functions. The authors introduce and deploy a suite of harmonic-analysis tools in mixed Lebesgue spaces, including a Fourier multiplier transference principle, vector-valued Bernstein inequalities, and vector-valued Littlewood–Paley theory for operator densities, enabling a duality-based approach to the orthonormal setting. They obtain local well-posedness for nonlinear Schrödinger equations with partially regular data and non-gauge nonlinearities, and establish well-posedness for the Hartree equation describing infinitely many fermions in Schatten spaces, with densities possessing mixed-norm regularity. These results advance dispersive PDE analysis on compact manifolds by combining partial-regularity ideas with mixed-norm and operator-density techniques, with potential implications for many-body quantum dynamics on tori.

Abstract

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schrödinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schrödinger equation has better regularity in mixed Lebesgue spaces. This complements the well-established theory over the past few decades, where initial data comes from the Sobolev space with respect to all spatial variables. As an application, we obtain local well-posedness for non-gauge-invariant nonlinearities with partially regular initial data. On the other hand, we extend refined Strichartz estimates for infinite systems of orthonormal functions, which generalizes the classical orthonormal Strichartz estimates on the torus by Nakamura [41] . As an application, we establish well-posedness for the Hartree equation for infinitely many fermions in some Schatten spaces. In the process, we develop several harmonic analysis tools for mixed Lebesgue spaces, e.g. Fourier multiplier transference principle, vector-valued Bernstein inequality, and vector-valued Littlewood--Paley theory for densities of operators, which may be of independent interest and complement the results of [45,55].

Refined Strichartz estimates and their orthornomal counterparts for Schrödinger equations on torus

TL;DR

This work develops refined Strichartz estimates and their orthonormal counterparts for the Schrödinger equation on the torus under a partial regularity framework, proving improved mixed Lebesgue regularity for initial data in the spaces and extending to infinite systems of orthonormal functions. The authors introduce and deploy a suite of harmonic-analysis tools in mixed Lebesgue spaces, including a Fourier multiplier transference principle, vector-valued Bernstein inequalities, and vector-valued Littlewood–Paley theory for operator densities, enabling a duality-based approach to the orthonormal setting. They obtain local well-posedness for nonlinear Schrödinger equations with partially regular data and non-gauge nonlinearities, and establish well-posedness for the Hartree equation describing infinitely many fermions in Schatten spaces, with densities possessing mixed-norm regularity. These results advance dispersive PDE analysis on compact manifolds by combining partial-regularity ideas with mixed-norm and operator-density techniques, with potential implications for many-body quantum dynamics on tori.

Abstract

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schrödinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schrödinger equation has better regularity in mixed Lebesgue spaces. This complements the well-established theory over the past few decades, where initial data comes from the Sobolev space with respect to all spatial variables. As an application, we obtain local well-posedness for non-gauge-invariant nonlinearities with partially regular initial data. On the other hand, we extend refined Strichartz estimates for infinite systems of orthonormal functions, which generalizes the classical orthonormal Strichartz estimates on the torus by Nakamura [41] . As an application, we establish well-posedness for the Hartree equation for infinitely many fermions in some Schatten spaces. In the process, we develop several harmonic analysis tools for mixed Lebesgue spaces, e.g. Fourier multiplier transference principle, vector-valued Bernstein inequality, and vector-valued Littlewood--Paley theory for densities of operators, which may be of independent interest and complement the results of [45,55].
Paper Structure (20 sections, 29 theorems, 267 equations, 2 figures)

This paper contains 20 sections, 29 theorems, 267 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Strichartz estimates
  3. Orthonormal Strichartz estimates
  4. Statement of the main results
  5. Applications to well-posedness
  6. Hartree equations for infinitely many particles
  7. Proof techniques and novelties
  8. Preliminaries
  9. Mixed Besov-type spaces
  10. Schatten spaces $\mathfrak{S}^{\alpha}$
  11. Duality principle in $\mathfrak{S}^{\alpha}$
  12. Complex interpolation in $\mathfrak{S}^{\alpha}$
  13. Harmonic analysis tools
  14. Littlewood-Paley theorem on mixed Lebesgue space
  15. Vector-valued Bernstein inequality
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $d \geq 1, 1 \leq k \leq d, 2 \leq \widetilde{r} \leq r<\infty,$ and $2<q \leq \infty$. If $q, r, \tilde{r}$ satisfy then for $f\in {H}^{\frac{1}{q}}(\mathbb{T}^d)$, we have

Figures (2)

  • Figure 1: Complete region for Strichartz estimate on $\mathbb{T}^d, d \geq 3.$
  • Figure 2: The shaded region represent $2 \leq \tilde{r} \leq r \leq \infty.$

Theorems & Definitions (53)

  • Theorem 2.1: Strichartz estimates
  • Corollary 2.2
  • Remark 2.3
  • Theorem 2.4: Orthonormal Strichartz estimates
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7: local well-posedness
  • Corollary 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 43 more