Differential and Integral Equations

On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise

Igor Kukavica, Kerem Uğurlu, and Mohammed Ziane

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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.

Article information

Source
Differential Integral Equations Volume 31, Number 3/4 (2018), 173-186.

Dates
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1513652422

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 60H15: Stochastic partial differential equations [See also 35R60] 60H30: Applications of stochastic analysis (to PDE, etc.)

Citation

Kukavica, Igor; Uğurlu, Kerem; Ziane, Mohammed. On the Galerkin approximation and strong norm bounds for the stochastic Navier-Stokes equations with multiplicative noise. Differential Integral Equations 31 (2018), no. 3/4, 173--186. https://projecteuclid.org/euclid.die/1513652422


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