Both $P_m$ and the <Leray-Helmholtz projection> are <orthogonal projections>, hence contractions in $L^2$. Taking the $H$ norm of the first Galerkin equation gives
$$
\nu|Au_m|
=\alpha|P_mP_\sigma(\theta_me_3)|
\leq\alpha\|\theta_me_3\|_2
=\alpha\|\theta_m\|_2.
$$
Therefore
$$
\boxed{|Au_m|\leq\frac{\alpha}{\nu}\|\theta_m\|_2}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-359/1/b/i/solution