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

Expand $P_{N}w$ in the orthonormal Stokes operator eigenbasis:The $H$ and $V$ norms satisfySince ${\lambda }_k\le {\lambda }_N$ on the projected space,This is the basic inverse estimate for a spectral projection of the Stokes operator.