Lieb--Thirring inequalities on the sphere

Abstract

We prove on the sphere ${\mathbb{S}}^2$ the Lieb--Thirring inequalities for orthonormal families of scalar and vector functions both on the whole sphere and on proper domains on ${\mathbb{S}}^2$. By way of applications we obtain an explicit estimate for the dimension of the attractor of the Navier--Stokes system on a domain on the sphere with Dirichlet non-slip boundary conditions.