The Navier-Stokes equations arise in fluid mechanics. The deterministic Navier-Stokes equations express the conservation of the momentum and mass for a Newtonian flow. The stochastic term accounts for the perturbation during the flow evolution. We considered the stochastic Navier-Stokes equations on the three-dimensional torus and the whole space $R^3$, aiming to see if there exists a unique global or local solution.
In this presentation, I will talk about the solvability of the stochastic Navier-Stokes equations with a multiplicative It\^o noise in $L^p$ spaces, where $p$ is greater than the spatial dimension 3. I will show the analytic tools we used in various domains to linearize the convective term and explain the arguments we exploited to show the nondegeneracy of stopping times. We obtained global existence with a large probability for $p>5$ and local existence almost surely for $p>3$. This presentation is based on joint works with I. Kukavica, F. Wang, and M. Ziane.