In this paper, we give a new characterization of the incompressible Navier--Stokes equations on a compact Riemannian manifold M via the Bellman dynamic programming principle on SG = SDiff(M), the group of volume preserving diffeomorphisms on M. The main result of this paper indicates the interesting relationships among the incompressible Navier--Stokes equations on M, the Hamilton-Jacobi-Bellman equation, and the viscous Burgers equation on SG = SDiff(M). In particular, we derive the incompressible Navier--Stokes equations and the revised incompressible Navier--Stokes equations on the two-dimensional torus. This extends Arnold's famous theorem on the geometric interpretation of the incompressible Euler equation to the incompressible Navier--Stokes equations on compact Riemannian manifolds.

Publication:

SIAM JOURNAL ON CONTROL AND OPTIMIZATION

http://dx.doi.org/10.1137/24M1658103

Author:

Li, Xiang-Dong(corresponding author)

Chinese Acad Sci, Acad Math & Syst Sci, State Key Lab Math Sci, 55 Zhongguancun East Rd, Beijing 100190, Peoples R China

Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China

Email address: xdli@amt.ac.cn

Liu, Guoping

Huazhong Univ Sci & Technol, Sch Math & Stat, 1037 Luoyu Rd, Wuhan 430074, Peoples R China

Email address: liuguoping@hust.edu.cn