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

TL;DR

The paper derives the incompressible Navier-Stokes equations on a compact Riemannian manifold $M$ by applying the Bellman dynamic programming principle to the infinite-dimensional group of volume-preserving diffeomorphisms $\mathrm{SDiff}(M)$. It establishes a Hamilton-Jacobi-Bellman framework on $M$ and on $SG$, connects the resulting equations to a viscous Burgers equation on $SG$, and shows how NS on $M$ can be reconstructed from SG data via push-forward through the evaluation map. In the zero-viscosity limit, it provides a deterministic DP approach yielding an Euler-type formulation with an external force. Overall, the work reveals a deep link between NS/Euler dynamics and HJB/Burgers equations on diffeomorphism groups, offering a novel geometric-probabilistic perspective on fluid equations on manifolds.

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 $SG={\rm SDiff}(M)$ 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 $SG={\rm SDiff}(M)$. 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 $SG={\rm SDiff}(M)$ of volume preserving diffeomorphisms.