Stochastic Navier--Stokes Equations for Turbulent Flows

Abstract

This paper concerns the fluid dynamics modelled by the stochastic flow \left\{ \begin{array}{l} \boldsymbol{\dot{\eta}}\left( t,x\right) =\boldsymbol{u}\left( t,\boldsymbol{\eta} \left( t,x\right) \right) +\boldsymbol{\sigma}\left( t,\boldsymbol{\eta}\left( t,x\right) \right) \circ\dot{W}, \\ \\ \boldsymbol{\eta}(0,x)=x, \end{array} \right. where the turbulent term is driven by the white noise $\dot{W}$. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components $\boldsymbol{u}(t,x)$ and $\boldsymbol{\sigma}(t,x)$ of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations. In the second part of the paper, we prove the existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in $W_{p}^{1}(\boldsymbol{R}^{d}),d >1,p > d. In the two-dimensional case, the existence and uniqueness of a global strong solution is shown. In the third part, we deal with the propagation of Wiener chaos by the stochastic Navier--Stokes equation and its relation to statistical moments of the solution.