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

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 $m$.

On ${\tilde{H}}_m$, use the inner productThe map in the question is continuous, and part a givesOn the sphere $\Vert v\Vert =\rho$ with any $\rho >R$,The Brouwer inward-pointing zero lemma therefore supplies $v^{\ast }$ in the ball with ${\Phi }_m(v^{\ast })=0$. Multiplication by $\nu \tilde{A}$ shows that this zero satisfiesThus every Galerkin system has at least one solution.