Lieb-Thirring inequalities on some manifolds

We prove Lieb-Thirring inequalities with improved constants on the two-dimensional sphere and the two-dimensional torus. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues

Sharp estimates for the number of degrees of freedom for the damped-driven 2D Navier--Stokes equations

We derive upper bounds for the number of asymptotic degrees (determining modes and nodes) of freedom for the two-dimensional Navier--Stokes system and Navier-Stokes system with damping. In the first case we obtain the previously known estimates in an explicit form, which are larger than the fractal dimension of the global attractor. However, for the Navier--Stokes system with damping our estimates for the number of the determining modes and nodes are comparable to the sharp estimates for the fractal dimension of the global attractor. Our investigation of the damped-driven 2D Navier--Stokes system is inspired by the Stommel--Charney barotropic model of ocean circulation where the damping represents the Rayleigh friction. We remark that our results equally apply to the Stommel--Charney model

On the Domain of Analyticity and Small Scales for the Solutions of the Damped-driven 2D Navier-Stokes Equations

We obtain a logarithmically sharp estimate for the space-analyticity radius of the solutions of the damped-driven 2D Navier-Stokes equations with periodic boundary conditions and relate this to the small scales in this system. This system is inspired by the Stommel--Charney barotropic ocean circulation model

One-dimensional interpolation inequalities, Carlson--Landau inequalities and magnetic Schrodinger operators

In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on multi-dimensional cylinders.Comment: 33