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

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The Stokes operator is $A=-P_\sigma\Delta$ on divergence-free vector fields, where $P_\sigma$ is the <Leray-Helmholtz projection>. On a periodic domain it equals $-\Delta$ on its domain, and its inverse gains two <Sobolev space>[Sobolev derivatives].

Stokes operator

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The Leray-Helmholtz projection $P_\sigma$ is the orthogonal projection of square-integrable vector fields onto divergence-free vector fields. Applying it to an incompressible momentum equation removes the pressure gradient.


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 Incoming links: Leray-Helmholtz projection