Global existence for perturbations of the 2D stochastic Navier–Stokes equations with space-time white noise

Abstract

We prove global in time well-posedness for perturbations of the 2D stochastic Navier–Stokes equations

$$\begin{aligned} \partial _t u + u \cdot \nabla u= & {} \Delta u - \nabla p + \zeta + \xi \;, \qquad u (0, \cdot ) = u_{0} \;,\\ {\text {div}}(u)= & {} 0 \;, \end{aligned}$$

driven by additive space-time white noise \( \xi \), with perturbation \( \zeta \) in the Hölder–Besov space \(\mathcal {C}^{-2 + 3\kappa } \), periodic boundary conditions and initial condition \( u_{0} \in \mathcal {C}^{-1 + \kappa } \) for any \( \kappa >0 \). The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a \( \log \)–correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation \( \zeta \) is not restricted to the Cameron–Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data \( u_{0}\) in \( L^{2} \), the critical space of initial conditions.