Studying Navier-Stokes equations using differential geometry

Question

I study Navier-Stokes in $\mathbb{R}$. But I am interested in applying Differential Geometry for these equations. If I extend my domain to a torus, would this enable me to use DG?

Answer 1

You can use differential geometry even if the domain is the usual open subset of $\mathbb R^d$ with compact, smooth boundary.

If you consider the measure preserving diffeomorphisms on a domain $\Omega$, consider it as some kind of infinite dimensional Lie group, consider the Lie algebra to be the divergence free vector fields on $\Omega$ with slip boundary conditions, put a "Riemannian metric" on the Lie algebra equal to the $L_2$ norm of the vector field, then the geodesics on this Lie group turn out to be precisely the solutions to the Euler equation.

This is described in one of the appendices of V.I. Arnold's book "Mathematical Methods of Classical Mechanics," and also in the book by V.I. Arnold and Boris A. Khesin "Topological Methods in Hydrodynamics." But I must admit I found these books rather hard to read.

I believe the manipulations are formal, and not rigorous. There is a paper by Ebin and Marsden that puts it into a more rigorous formulation, but I don't think that they rigorously prove the geodesics are solutions to the Euler equation - instead I think they show that the Hamiltonian equations one gets by formally manipulating the Euler-Lagrange equations can be described rigorously.