The formulation of the Navier-Stokes equations on the Riemannian manifolds - Magdalena Czubak

Workshop on Geometry and Analysis of Fluid Flows with a Special Tribute to David Ebin
January 2023

Magdalena Czubak, University of Colorado Boulder

Abstract: In their 1970 seminal article, David Ebin and Jerrold Marsden presented the first in-depth study of the Navier-Stokes equations on the Riemannian manifolds. There are several inequivalent formulations of the Navier-Stokes equations on the Riemannian manifolds.
This is due to the different possibilities for the Laplacian operator acting on vector fields. In their article, Ebin and Marsden indicated what form the Navier-Stokes equations should take when considered on a Riemannian manifold. In this talk, we discuss the operator proposed by Ebin and Marsden, and present the recent developments in the study of the formulation of the Navier-Stokes equations in the setting of Riemannian manifolds.