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Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

Sonae Hadama, Andrew Rout

Abstract

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schrödinger equations (NLSS) on a circle. We show that this renormalised density satisfies better orthonormal Strichartz estimates than the non-renormalised density, which was considered in Nakamura (2020). We leave as a conjecture the optimal range of exponents for these Strichartz estimates. As an application, we determine the critical Schatten exponent below which the cubic renormalised NLSS on the circle is globally well-posed and above which it is ill-posed. Finally, we show that the improvement for orthonormal Strichartz estimates satisfied by the renormalised density on $\mathbb{T}^d$ for $d \geq 2$ is minimal.

Applications of renormalisation to orthonormal Strichartz estimates and the NLS system on the circle

Abstract

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schrödinger equations (NLSS) on a circle. We show that this renormalised density satisfies better orthonormal Strichartz estimates than the non-renormalised density, which was considered in Nakamura (2020). We leave as a conjecture the optimal range of exponents for these Strichartz estimates. As an application, we determine the critical Schatten exponent below which the cubic renormalised NLSS on the circle is globally well-posed and above which it is ill-posed. Finally, we show that the improvement for orthonormal Strichartz estimates satisfied by the renormalised density on for is minimal.

Paper Structure

This paper contains 22 sections, 29 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Nonlinear Schrödinger system
  3. Notation
  4. Orthonormal Strichartz estimates
  5. Orthonormal Strichartz estimates for density function
  6. Orthonormal Strichartz estimates for the renormalised density
  7. Higher dimensional case
  8. Applications to NLS systems
  9. Method of proof
  10. Outline of the paper
  11. Preliminary results
  12. Orthonormal Strichartz estimates
  13. Proofs of Propositions \ref{['prop:L2 OStri 1D torus paraphrase']} and \ref{['thm:Nakamura paraphrase']}
  14. Proof of Theorem \ref{['thm:L2 bound renormalised']}
  15. Proof of Theorem \ref{['thm:L3 bound renormalised']}
  16. ...and 7 more sections

Key Result

Proposition 1.5

If $\alpha=1$, then we have for any $\gamma_0 \in \mathfrak{S}^\alpha$. Moreover, this is sharp in the sense that eq:L2 OStri 1D torus paraphrase fails if $\alpha > 1$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (77)

  • Definition 1.1: Density function
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5: Orthonormal version of \ref{['eq:L4']}
  • Proposition 1.6: Orthonormal version of \ref{['eq:L6']}
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 1.10: Renormalised density function
  • ...and 67 more