Vorticity, Helicity, Intrinsinc geometry for Navier-Stokes equations

TL;DR

This paper explores the intrinsic geometric properties of Navier-Stokes equations on Riemannian manifolds, introducing new connections that relate vorticity and helicity via the Ricci tensor in three dimensions.

Contribution

It introduces a novel family of connections linked to Navier-Stokes solutions, connecting vorticity and helicity through the Ricci tensor intrinsically in 3D.

Findings

Established a geometric framework relating vorticity and helicity

Linked solutions of Navier-Stokes to Ricci curvature in a new way

Provided insights into the intrinsic geometry of fluid dynamics

Abstract

We will consider the Navier-Stokes equation on a Riemannian manifold M with Ricci tensor bounded below, the involved Laplacian operator is De Rham-Hodge Laplacian. The novelty of this work is to introduce a family of connections which are related to solutions of the Navier-Stokes equation, so that vorticity and helicity can be linked through the associated time-dependent Ricci tensor in intrinsic way in the case where dim(M) = 3. MSC 2010: 35Q30, 58J65