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

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). 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 $d\ge 2$ is minimal.