| Abstract: |
| In any bounded domain $\Omega \subset \RR^3$, $ \partial \Omega \in C^2$, let $A=-P\Delta$ denote the Stokes operator, and $v \in C^0([o, a], D_{A^\beta})$ any function being bounded by constant $c_v\geq ||A^\beta v(t)||_{L^2(\Omega)}$ uniformly in $t\in [o, a]$, with some $\beta \in (0, 1]$. For the approximations
$$
v_k(t) = P_k v(t) = \sum^k_{j=1} < v (t), e_j > e_j
$$
in terms of the complete orthonormal system of eigenfunctions $e_j$ of $A$ in $L^2_\sigma (\Omega)$, $ Ae_j = \lambda_j e_j$, $ 0 < \lambda_j \to \infty$ if $j \to \infty$, a short proof gives the error estimate
$$
||A^\alpha (v(t) - v_k(t)) ||_{L^2(\Omega)} \leq \frac{C_v}{\lambda^{\beta - \alpha}}
$$
for all $\lambda \in [0, \beta)$ uniformly in $t\in [o, a]$.
\bibitem Roger Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North Holland, Amsterdam 1979. |
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