Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / b

On the finite-dimensional space $H_m$, the Galerkin equation is an autonomous system of ordinary differential equations whose right-hand side is a quadratic polynomial in the coefficients of $u_m$. It is locally Lipschitz, so the Picard-Lindelof theorem gives a unique maximal local solution.

Taking the $H$ inner product with $u_m$ givesThe advecting field $P_N{u}_m$ is divergence free. Periodicity and the skew-symmetry of incompressible transport therefore make the nonlinear term zero. HenceA finite-dimensional solution can cease to exist only if its norm diverges. This uniform bound prevents such blow-up, so the solution extends uniquely through every interval $[0,T]$.

Integrating the exact Galerkin energy identity givesThus one may takefor the first two requested bounds; these constants happen not to grow with $T$.

The Galerkin equation and contractivity of $P_m$ on $V^{\prime }$ giveSince $\Vert A{u}_m{\Vert }_{V^{\prime }}=\Vert u_m\Vert$, part a givesConsequentlyAll three constants are independent of $m$.