Navier-Stokes equations in riemannian geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:36:56Z http://mathoverflow.net/feeds/question/76325 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry Navier-Stokes equations in riemannian geometry francis-jamet 2011-09-25T12:48:35Z 2011-10-31T20:26:30Z <p>Hello,</p> <p>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 $u$. </p> <p>The problem is due to Antoine Balan.</p> <p>Do you have references ?</p> <p>Thanks in advance.</p> http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry/76329#76329 Answer by Giuseppe for Navier-Stokes equations in riemannian geometry Giuseppe 2011-09-25T14:50:32Z 2011-09-25T16:42:56Z <p>You could look at the paper: <a href="http://www.jstor.org/pss/1970699" rel="nofollow">Groups of Diffeomorphisms and the motion of an incompressible fluid, by Ebin and Marsden</a>.</p> <p>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.</p> <p>Ebin and Marsden promptly employed this reformulation to obtain existence and uniqueness results for these equations on compact oriented riemannian manifolds.</p> <p>This circle of ideas is one of the first important application of infinite dimensional manifolds as remarked by Stephen Smale.</p> <hr> <p>By the way, should not the equation contain the time derivative of the unknown $u$?</p> http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry/76346#76346 Answer by Denis Serre for Navier-Stokes equations in riemannian geometry Denis Serre 2011-09-25T16:50:41Z 2011-09-25T16:50:41Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry/76390#76390 Answer by Richard Montgomery for Navier-Stokes equations in riemannian geometry Richard Montgomery 2011-09-26T04:19:38Z 2011-09-26T04:19:38Z <p>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!</p> http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry/76452#76452 Answer by YangMills for Navier-Stokes equations in riemannian geometry YangMills 2011-09-26T20:59:11Z 2011-09-26T20:59:11Z <p>For what it's worth, the Navier-Stokes equation on manifolds is also mentioned in this recent paper <a href="http://arxiv.org/pdf/1107.2698" rel="nofollow">http://arxiv.org/pdf/1107.2698</a>, see (1.16) there, in connection with another flow for vector fields that the authors define.</p> http://mathoverflow.net/questions/76325/navier-stokes-equations-in-riemannian-geometry/79650#79650 Answer by Claire for Navier-Stokes equations in riemannian geometry Claire 2011-10-31T20:26:30Z 2011-10-31T20:26:30Z <p>Do you think it would be possible to extend the results of Arnol'd to the Navier-Stokes equation ?</p>