Stochastic vorticity equation in $\mathbb R^2$ with not regular noise

TL;DR

This paper proves the existence of unique strong solutions for the 2D stochastic vorticity form of Navier-Stokes equations driven by irregular multiplicative white noise, advancing understanding of stochastic fluid dynamics.

Contribution

It establishes the existence and uniqueness of strong solutions for 2D stochastic Navier-Stokes equations with irregular multiplicative noise, where traditional Itô calculus methods are not applicable.

Findings

Existence of unique strong solutions in 2D stochastic vorticity equations.

Handling of irregular multiplicative noise without standard Itô calculus.

Advancement in stochastic fluid dynamics theory.

Abstract

We consider the Navier-Stokes equations in vorticity form in $\mathbb{R}^2$ with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the It\^o calculus in $L^q$ spaces, $1<q<\infty$. We prove the existence of a unique strong (in the probability sense) solution.