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 ξ_0 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.

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

Ferrario B.;Zanella M.
2019-01-01

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 ξ_0 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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/1268886
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 6
  • ???jsp.display-item.citation.isi??? 7
social impact