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

• ### Alexei Ilyin

Keldysh Institute of Applied Mathematics, Moscow, Russia; Sirius University of Science and Technology, Sochi, Russia
• ### Anna Kostianko

Imperial College London, United Kingdom
• ### Sergey Zelik

University of Surrey, Guildford, UK; Lanzhou University, China, Keldysh Institute of Applied Mathematics, Moscow, Russia

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## 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 two-dimensional torus are also given.