Stochastic vorticity equation in mathbb R² with not regular noise

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.