Lieb–Thirring integral inequalities and their applications to attractors of the Navier–Stokes equations

Integral inequalities of Lieb–Thirring type and their generalizations are proved. All the corresponding constants are given in explicit form. Special attention is devoted to applications to the attractors of the two-dimensional Navier–Stokes equations. In particular, an explicit two-sided estimate of the attractor dimension is established for the Kolmogorov problem on the two-dimensional elongated torus.