Global existence for the stochastic Navier–Stokes equations with small $$L^{p}$$ data

We consider the stochastic Navier–Stokes equations in $${\mathbb T}^{3}$$ with multiplicative white noise. We construct a unique local strong solution with initial data in $$L^p$$ , where $$p>5$$ . We also address the global existence of the solution when the initial data is small in $$L^p$$ , with the same range of p.