Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / b / i / Solution

Take the $\tilde{H}$ inner product of the finite-dimensional equation with $u_m$. Part a gives exact cancellation of the nonlinear term:By the Cauchy-Schwarz inequality and the Poincare inequality,Thus every solution satisfiesThe bound is independent of $m$.