| Abstract: |
| In this talk, we use the Lagrange multiplier method to derive the incompressible Euler and Navier-Stokes equations on a compact Riemannian manifold $M$, in which the pressure is given by a variant of the Lagrange multiplier for the incompressible condition ${\rm{div}}~u=0$.
Moreover, we give a new derivation of the incompressible
Navier-Stokes equation on a compact Riemannian manifold $M$ via the Bellman dynamic programming principle on the infinite dimensional group of diffeomorphisms $G={\rm Diff}(M)$. In particular,
in the inviscid case, we give a new derivation of the incompressible Euler equation on a compact Riemannian manifold $M$. Our method provides an explicit construction of a solution to the incompressible Euler and Navier-Stokes equations via the value function and the Lagrange multiplier of a deterministic or stochastic optimal control problem on $G={\rm Diff}(M)$. Joint work with Guoping Liu (HUST, Wuhan). |
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