Expand $P_Nw$ in the orthonormal <Stokes operator> eigenbasis:
$$
P_Nw=\sum_{k=1}^Nw_k\phi_k.
$$
The $H$ and $V$ norms satisfy
$$
|P_Nw|^2=\sum_{k=1}^N|w_k|^2,
\qquad
\|P_Nw\|^2=(AP_Nw,P_Nw)
=\sum_{k=1}^N\lambda_k|w_k|^2.
$$
Since $\lambda_k\leq\lambda_N$ on the projected space,
$$
\boxed{\|P_Nw\|\leq\lambda_N^{1/2}|P_Nw|}.
$$
This is the basic inverse estimate for a <spectral projection of the Stokes operator>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / a / i / Solution

{title2=$P_N$}

The spectral projection $P_N$ retains the first $N$ Stokes eigenfunctions. On its finite-dimensional range, one derivative costs at most $\lambda_N^{1/2}$ and all Sobolev norms are equivalent.


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 Incoming links: Spectral projection of the Stokes operator