We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when t
he forcing term belongs to the whole scale of homogeneous Sobolev spaces from -1 to 1
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 ine
qualities for magnetic Schrodinger operators on multi-dimensional cylinders.
We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method
is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlsons inequalities and spectral theory of discrete operators are given.
