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I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations which "embed" this manifold in $R^3$?

This would be somewhat analogous to the way that the Poincare Disk Model transforms straight lines in the hyperbolic plane into arcs of circles that are orthogonal to the boundary circle. Except in this case, the "straight lines" are transformed into fluid particle trajectories in $R^3$.

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Navier-Stokes equation if solved, does not emit a solution that describes particle trajectories as other classical mechanics does. The solution would represent a velocity field instead. I do not think that Navier-Stokes equations even require it makes sense to talk about the particle trajectory but if one does, it seems natural that what you are asking is to add yet one more equation to the ensemble and constrain that as well. – marshal craft Jul 5 '15 at 5:42
@marshalcraft: I used the word "fluid particle" to mean the quasi-particle associated with the velocity field. I think you are missing the point of this question, which is that if you could show that the N-S equations are the embedding of a manifold, it would create a clear strategy for showing that they do not possess smooth solutions. See Willie Wong's answer to my question here – Matt Calhoun Jul 6 '15 at 14:48
up vote 14 down vote accepted

For the Navier Stokes equation, I am rather doubtful, because the presence of the viscosity term makes the equation not time-symmetric. And the geodesic flow is necessarily time-symmetric.

If you remove the viscosity term, you are down to the Euler equation, for which there is a well-known characterisation due to V I Arnold.

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Maybe Lagrangian coordinates is what you are after.

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