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 Advances in Differential Equations 14 (2009), 567–600, regarding moments of strong pathwise solutions to the Navier–Stokes equations in a two-dimensional bounded domain 𝒪. We prove that Eφ(‖u(t)‖_H¹(𝒪)²)<∞ 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 Eφ_ε(‖u(t)‖_H¹(𝒪0)²)<∞, where φ_ε(x)=(1+x)^(1−ε)/2, for any deterministic t>0 and any ε>0.