Generalized stochastic Lagrangian paths for the Navier-Stokes equation

• Marc Arnaudon ^[1] ; Ana Bela Cruzeiro ^[3] ; Shizan Fang ^[2]
  1. [1] University of Bordeaux

     University of Bordeaux

     Arrondissement de Bordeaux, Francia

     
  2. [2] University of Burgundy

     University of Burgundy

     Arrondissement de Dijon, Francia

     
  3. [3] Instituto Superior Técnico Lisboa, Portugal

  Mostrar afiliaciones +
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 18, Nº 3, 2018, págs. 1033-1060
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In the note added in proof of the seminal paper [14], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In the spirit of Brenier’s generalized flows for the Euler equation, we introduce a class of semimartingales on a compact Riemannian manifold. We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden’s Laplacian. We also show that for the case of torus, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.