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

We investigate the convergence of the Galerkin approximation 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} \left[ \sup_{t \in [0,T]} \phi_1(\lVert (u(t)-u^n(t)) \rVert^2_V) \right] \rightarrow 0 \end{equation*} as $n \rightarrow \infty$ for any deterministic time $T > 0$ and for a specified moment function $\phi_1(x)$ where $u^n(t,x)$ denotes the Galerkin approximation of the solution $u(t,x)$. 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 approximation up to a deterministic time $T$ when the $V$-norm is replaced by the $H$-norm.