Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

Abstract: The aim of this work is to prove an existence and uniqueness result of Kato-Fujita type for the Navier-Stokes equations, in vorticity form, in $2-D$ and $3-D$, perturbed by a gradient type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by V. Barbu and the second named author in \cite{b1}, that treats the stochastic $3D$-Navier-Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in \cite{b1}, but the transformation is different and adapted to our gradient type noise. Then global unique existence results are proved for the transformed equation, while for the original stochastic Navier-Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.