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

Igor Kukavica; Vlad Vicol

doi:10.3233/ASY-141237

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

Igor Kukavica, Vlad Vicol

Mathematics

Research output: Contribution to journal › Article › peer-review

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. 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 Eφ_O(u(t)H¹(₀)²)<, where φ_O(x)=(1+x)¹², for any deterministic t>0 and any ε>0.

Original language	English (US)
Pages (from-to)	189-206
Number of pages	18
Journal	Asymptotic Analysis
Volume	90
Issue number	3-4
DOIs	https://doi.org/10.3233/ASY-141237
State	Published - 2014

Keywords

moments
pathwise solutions
stochastic Navier-Stokes equations

ASJC Scopus subject areas

General Mathematics

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10.3233/ASY-141237

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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. 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.",

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AU - Kukavica, Igor

AU - Vicol, Vlad

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N2 - 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.

AB - 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.

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KW - pathwise solutions

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

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