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

Alexei Ilyin; Anna Kostianko; Sergey Zelik

doi:10.4171/90-1

BooksStandalone TitlesCollected Volumepp. 583–608

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 $α$ -models approximating them. New applications to interpolation inequalities on the two-dimensional torus are also given.