Lieb--Thirring inequalities on the sphere

TL;DR

This paper establishes Lieb--Thirring inequalities on the sphere, providing explicit estimates for attractor dimensions in Navier--Stokes systems with boundary conditions, advancing mathematical understanding of spectral inequalities in spherical domains.

Contribution

It extends Lieb--Thirring inequalities to the sphere and applies these results to estimate the attractor dimension for Navier--Stokes equations on spherical domains.

Findings

Lieb--Thirring inequalities are proven on $ ext{S}^2$ for scalar and vector functions.

Explicit estimates for the attractor dimension of Navier--Stokes systems on the sphere are derived.

The results apply to both the whole sphere and proper subdomains with boundary conditions.

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.