Applications of the Lieb--Thirring and other bounds for orthonormal systems in mathematical hydrodynamics

TL;DR

This paper explores bounds for orthonormal systems in mathematical hydrodynamics, focusing on their role in estimating the dimension of attractors in Navier–Stokes equations and related models, with new applications to interpolation inequalities.

Contribution

It introduces new bounds and applications for orthonormal systems in hydrodynamics, improving estimates for attractor dimensions and interpolation inequalities.

Findings

Derived optimal bounds for global attractor dimensions.

Established new interpolation inequalities on the 2D torus.

Enhanced understanding of orthonormal system estimates in fluid dynamics.

Abstract

We discuss the estimates for the $L^p$-norms of systems of functions that are orthonormal in $L^2$ and $H^1$, respectively, and their essential role in deriving good or even optimal bounds for the dimension of global attractors for the classical Navier--Stokes equations and for a class of $\alpha$-models approximating them. New applications to interpolation inequalities on the 2D torus are also given.