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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.15997 |
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| _version_ | 1866910876246736896 |
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| author | Li, Xiang-Dong Liu, Guoping |
| author_facet | Li, Xiang-Dong Liu, Guoping |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Navier-Stokes equations and the Hamilton-Jacobi-Bellman equation on the group of volume preserving diffeomorphisms Li, Xiang-Dong Liu, Guoping Probability 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. |
| title | On the Navier-Stokes equations and the Hamilton-Jacobi-Bellman equation on the group of volume preserving diffeomorphisms |
| topic | Probability |
| url | https://arxiv.org/abs/2403.15997 |