Navier-Stokes equations in riemannian geometry - MathOverflow

Navier-Stokes equations in riemannian geometry

2011-09-25T12:48:35Z

Hello,

The Navier-Stokes equations can be written on a riemannian manifold: $$\dot{u}+\nabla_u u+ \Delta u=(df)^* $$ $$d^* u=0$$ where $\nabla$ is the Levi-Civita connection, $u$ is a vector fields, $\Delta$ is the laplacian, $df$ is the differential of $f$, $(df)^* $ is the dual of $df$ by the metric, $d^*u$ is the divergence of $u$.

The problem is due to Antoine Balan.

Do you have references ?

Thanks in advance.

Answer by Giuseppe for Navier-Stokes equations in riemannian geometry

2011-09-25T14:50:32Z

You could look at the paper: Groups of Diffeomorphisms and the motion of an incompressible fluid, by Ebin and Marsden.

About two centuries after Euler, in 1966 Arnold gave a geometric reformulation of the classical equations for an imcompressible fluid in terms of the geodesic spray of left invariant metric on an infinite dimensional Lie Group.

Ebin and Marsden promptly employed this reformulation to obtain existence and uniqueness results for these equations on compact oriented riemannian manifolds.

This circle of ideas is one of the first important application of infinite dimensional manifolds as remarked by Stephen Smale.

By the way, should not the equation contain the time derivative of the unknown $u$?

Answer by Denis Serre for Navier-Stokes equations in riemannian geometry

2011-09-25T16:50:41Z

The answer and comments about Arnold and Marsden papers are a little off side. They concern the equation of inviscid fluids, called Euler equation. This differs from Navier-Stokes by the highest-order derivatives $\Delta u$. This changes completely the functional analysis background. Also, Euler equation has a geometrical interpretation (geodesics on the group of measure-preserving diffeomorphisms), whereas Navier-Stokes has not.

I am not aware of references for Navier-Stokes on manifolds. However, I don't think that this is a real problem. What has been important so far for Navier-Stokes is the space dimension and the embedding theorems we have between functional spaces like Sobolev, Besov and others. For instance, the Cauchy problem must be globally well-posed on every compact surface, and locally well-posed on $3$-manifolds.

Answer by Richard Montgomery for Navier-Stokes equations in riemannian geometry

2011-09-26T04:19:38Z

You are missing a $\dot u$ in your equation! We want a dynamic vector field. The sign of your $\nabla_u u$ and $\Delta u$ are usually taken to be opposite, as with the sign of your $df^*$ and $\nabla_u u$. See p. 63 of Arnol'd-Khesin's book `Topological Methods in Fluid Mechanics'. Arnol'd and Khesin definitely knew how to do this. Khesin is still alive!

Answer by YangMills for Navier-Stokes equations in riemannian geometry

2011-09-26T20:59:11Z

For what it's worth, the Navier-Stokes equation on manifolds is also mentioned in this recent paper http://arxiv.org/pdf/1107.2698, see (1.16) there, in connection with another flow for vector fields that the authors define.

Answer by Claire for Navier-Stokes equations in riemannian geometry

2011-10-31T20:26:30Z

Do you think it would be possible to extend the results of Arnol'd to the Navier-Stokes equation ?