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March/April 2018 On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise
Kerem Uğurlu, Mohammed Ziane, Igor Kukavica
Differential Integral Equations 31(3/4): 173-186 (March/April 2018).

Abstract

We investigate the convergence of the Galerkin approximations for the stochastic Navier-Stokes equations in an open bounded domain $\mathcal{O}$ with the non-slip boundary condition. We prove that \begin{equation*} \mathbb{E} \Big [ \sup_{t \in [0,T]} \phi_1(\lVert (u(t)-u^n(t)) \rVert^2_V) \Big ] \rightarrow 0, \end{equation*} as $n \rightarrow \infty$ for any deterministic time $T > 0$ and for a specified moment function $\phi_1$ where $u^n(t)$ denotes the Galerkin approximations of the solution $u(t)$. Also, we provide a result on uniform boundedness of the moment $\mathbb{E} [ \sup_{t \in [0,T]} \phi(\lVert u(t) \rVert^2_V) ] $ where $\phi$ grows as a single logarithm at infinity. Finally, we summarize results on convergence of the Galerkin approximations up to a deterministic time $T$ when the $V$-norm is replaced by the $H$-norm.

Citation

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Kerem Uğurlu. Mohammed Ziane. Igor Kukavica. "On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise." Differential Integral Equations 31 (3/4) 173 - 186, March/April 2018.

Information

Published: March/April 2018
First available in Project Euclid: 19 December 2017

zbMATH: 06837093
MathSciNet: MR3738194

Subjects:
Primary: 35Q30, 60H15, 60H30

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.31 • No. 3/4 • March/April 2018
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