Moments for strong solutions of the 2D stochastic Navier-Stokes equations in a bounded domain

We address a question posed by Glatt-Holtz and Ziane in [GHZ09, Remark 2.1 (ii)], regarding moments of strong pathwise solutions to the Navier-Stokes equations in a two-dimensional bounded domain O. We prove that Eφ(‖u(t)‖2H1(O)) < ∞ for any deterministic t > 0, where φ(x) = log(1 + log(1 + x)). Such moment bounds may be used to study statistical properties of the long time behavior of the equation. In addition, we obtain algebraic moment bounds on compact subdomains O0 of the form Eφε(‖u(t)‖2H1(O0)) < ∞ , where φε(x) = (1 + x) (1−ε)/2, for any deterministic t > 0 and any ε > 0.