Alexei Ilyin: Spectral inequalities and Navier-Stokes equations

Spectral inequalities are very important in finding physically relevant estimates for the number of the degrees of freedom of dissipative dynamical systems. In the context of the Navier-Stokes system we shall discuss two types of spectral inequalities, namely, Berezin-Li-Yau-type inequalities for the eigenvalues of elliptic operators and systems (including the Stokes system), and the Lieb-Thirring inequalities (including inequalities on some manifolds: periodic case, 2d sphere, 2d torus).