On the Navier-Stokes equations and the Hamilton-Jacobi-Bellman equation on the group of volume preserving diffeomorphisms

Abstract

In this paper, we give a new derivation of the incompressible Navier-Stokes equations on a compact Riemannian manifold $M$ via the Bellman dynamic programming principle on the infinite dimensional group $SG=\mathrm{SDiff}(M)$ of volume preserving diffeomorphisms. In particular, when the viscosity vanishes, we give a new derivation of the incompressible Euler equation on a compact Riemannian manifold. The main result of this paper indicates an interesting relationship among the incompressible Navier-Stokes equations on $M$, the Hamilton-Jacobi-Bellman equation and the viscous Burgers equation on $SG=\mathrm{SDiff}(M)$. This extends Arnold's famous theorem on the geometric interpretation of the incompressible Euler equation on a compact Riemannian manifold $M$ by the geodesic equation on the group $SG=\mathrm{SDiff}(M)$ of volume preserving diffeomorphisms.