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

Article type: Research Article

Authors: Kukavica, Igor^; | Vicol, Vlad

Affiliations: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA. E-mail: [email protected] | Department of Mathematics, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected]

Note: [] Corresponding author. E-mail: [email protected]

Abstract: 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 Eφε(‖u(t)‖H1(𝒪0)2)<∞, where φε(x)=(1+x)(1−ε)/2, for any deterministic t>0 and any ε>0.

Keywords: stochastic Navier–Stokes equations, moments, pathwise solutions

DOI: 10.3233/ASY-141237

Journal: Asymptotic Analysis, vol. 90, no. 3-4, pp. 189-206, 2014