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<p>In this paper, we use the Lagrange multiplier method to derive the incompressible Euler and Navier-Stokes equations on a compact Riemannian manifold <i>M</i>, in which the pressure is given by a variant of the Lagrange multiplier for the incompressibility condition div <i>u</i> = 0. Moreover, we give a new derivation of the incompressible Navier-Stokes equations on a compact Riemannian manifold <i>M</i> via the Bellman dynamic programming principle on the infinite-dimensional group <i>G</i> = Diff(<i>M</i>) of diffeomorphisms. In particular, in the inviscid case, we give a new derivation of the incompressible Euler equation on a compact Riemannian manifold <i>M</i>. 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 and stochastic optimal control problem on <i>G</i> = Diff(<i>M</i>).</p>

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On the Lagrange multiplier method to the Euler and Navier-Stokes equations on compact Riemannian manifolds

  • 期刊论文

  • 发表日期：

    2026年4月27日

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  • Xiang-Dong Li,
  • Guoping Liu

摘要

In this paper, 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 incompressibility condition div u = 0. Moreover, 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 G = Diff(M) of diffeomorphisms. 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 and stochastic optimal control problem on G = Diff(M).