  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.

Analysis of PDEs;Spectral Theory