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. We prove that Eφ(u(t)H1()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 0 of the form EφO(u(t)H1(0)2)<, where φO(x)=(1+x)12, for any deterministic t>0 and any ε>0.
All Science Journal Classification (ASJC) codes
- pathwise solutions
- stochastic Navier-Stokes equations