Navier-Stokes equations in Riemannian geometry

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

The problem is due to Antoine Balan.

Do you have references?

• Have you looked at the work of Marsden and Weinstein? Sep 25 '11 at 14:05
• Moreover, googling "Navier-Stokes Riemannian manifold" produces a lot of hits. Sep 25 '11 at 19:27
• Do you think it would be possible to extend the results of Arnol'd to the Navier-Stokes equation ?
– user18921
Oct 31 '11 at 20:26
• For instance, considering the Navier-Stokes equation as a small perturbation of the Euler equations, just as it was done by Cruzeiro et al., but on a stochastic point of view
– user18921
Oct 31 '11 at 20:34
• From the point of view of extending the results from flat space to compact manifolds, there is no difficulty at all: arxiv.org/abs/0901.4412 Apr 3 '12 at 22:55

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.

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!

• Dear Richard Montgomery, I would add a link to the text "Topological Methods in Hydrodynamics" by Arnol'd and Khesin. books.google.com/…
– agt
Sep 26 '11 at 5:24

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 $u$?

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.

I would write the Navier-Stokes equations on a Riemannian manifold $$(\mathcal M,g)$$ in a slightly different way. The unknown is still a time-dependent vector field $$v$$, to which you can associate a one-form $$u$$, defined in the charts by $$\langle u(x), T\rangle_{T_x^*(\mathcal M), T_x(\mathcal M)}=g(v(x),T), \quad \text{u=gv for short.}$$ Then the equation is $$\partial_t u+\mathcal L_v(u)+\nu d^* du=dq,\quad \text{div} v=0,$$ where $$\mathcal L_v$$ is the Lie derivative with respect to $$v$$. Note that, if $$\Omega_0$$ is an orientation of $$\mathcal M$$, we define the divergence of $$v$$ by the formula $$\mathcal L_v(\Omega_0)=(\text{div} v)\Omega_0.$$ We may define now the vorticity $$\omega$$ as $$du$$ and get the equations $$\partial_t \omega+\mathcal L_v(\omega)+\nu dd^* \omega=0,\quad \text{div} v=0, \omega=d(gv).$$