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Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space

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  • Delbaen, Freddy
  • Qiu, Jinniao
  • Tang, Shanjian

Abstract

A coupled forward–backward stochastic differential system (FBSDS) is formulated in spaces of fields for the incompressible Navier–Stokes equation in the whole space. It is shown to have a unique local solution, and further if either the Reynolds number is small or the dimension of the forward stochastic differential equation is equal to two, it can be shown to have a unique global solution. These results are shown with probabilistic arguments to imply the known existence and uniqueness results for the Navier–Stokes equation, and thus provide probabilistic formulas to the latter. Related results and the maximum principle are also addressed for partial differential equations (PDEs) of Burgers’ type. Moreover, from truncating the time interval of the above FBSDS, approximate solution is derived for the Navier–Stokes equation by a new class of FBSDSs and their associated PDEs; our probabilistic formula is also bridged to the probabilistic Lagrangian representations for the velocity field, given by Constantin and Iyer (2008) and Zhang (2010); finally, the solution of the Navier–Stokes equation is shown to be a critical point of controlled forward–backward stochastic differential equations.

Suggested Citation

  • Delbaen, Freddy & Qiu, Jinniao & Tang, Shanjian, 2015. "Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2516-2561.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2516-2561
    DOI: 10.1016/j.spa.2015.02.014
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    References listed on IDEAS

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    3. Cruzeiro, Ana Bela & Shamarova, Evelina, 2009. "Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4034-4060, December.
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