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

TL;DR

This paper develops a renormalisation method for the density in nonlinear Schrödinger systems on a circle, leading to improved orthonormal Strichartz estimates and insights into well-posedness thresholds.

Contribution

It introduces a renormalisation procedure that enhances Strichartz estimates for NLSS densities and determines critical Schatten exponents for global well-posedness.

Findings

Renormalised density satisfies better Strichartz estimates than non-renormalised.

Identifies the critical Schatten exponent for NLSS well-posedness.

Shows minimal improvement in Strichartz estimates for higher-dimensional tori.

Abstract

In this paper, we introduce a renormalisation procedure for the density associated with the system of nonlinear Schr\"odinger 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.