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

## TL;DR

This paper establishes the existence and uniqueness of solutions for 2D and 3D stochastic vorticity equations with gradient-dependent Gaussian noise, modeling turbulence, with global solutions in the transformed setting and local solutions in the original formulation.

## Contribution

It introduces a novel transformation for gradient-type noise in stochastic Navier-Stokes equations and proves global existence and uniqueness results in the vorticity form.

## Key findings

- Proved global unique existence for the transformed equations.
- Established local existence for the original stochastic equations.
- Extended previous results to gradient-dependent noise in turbulence modeling.

## 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.