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Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations

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  • Ferrario, Benedetta
  • Zanella, Margherita

Abstract

We consider the two dimensional Navier–Stokes equations in vorticity form with a stochastic forcing term given by a gaussian noise, white in time and colored in space. First, we prove existence and uniqueness of a weak (in the Walsh sense) solution process ξ and we show that, if the initial vorticity ξ0 is continuous in space, then there exists a space–time continuous version of the solution. In addition we show that the solution ξ(t,x) (evaluated at fixed points in time and space) is locally differentiable in the Malliavin calculus sense and that its image law is absolutely continuous with respect to the Lebesgue measure on R.

Suggested Citation

  • Ferrario, Benedetta & Zanella, Margherita, 2019. "Absolute continuity of the law for the two dimensional stochastic Navier–Stokes equations," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1568-1604.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:5:p:1568-1604
    DOI: 10.1016/j.spa.2018.05.015
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    References listed on IDEAS

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    1. Márquez-Carreras, D. & Mellouk, M. & Sarrà, M., 2001. "On stochastic partial differential equations with spatially correlated noise: smoothness of the law," Stochastic Processes and their Applications, Elsevier, vol. 93(2), pages 269-284, June.
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