Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 2 / a

Take the $L^2$ inner product of the first Galerkin equation with ${\omega }_m$. The orthogonal projection may be removed against ${\omega }_m\in {\tilde{H}}_m$, and skew-symmetry of incompressible transport cancels the nonlinear term. HenceThe Cauchy-Schwarz inequality and Young inequality giveThus both quantities requested in the first estimate are bounded by, for example,

Next take the inner product with $-\Delta {\omega }_m$. Periodicity and incompressibility giveThe given curl identity and the two-dimensional Gagliardo-Nirenberg inequality implyApplying Young's inequality to this term and to $(g,-\Delta {\omega }_m)$ yieldsThe first estimate bounds the right-hand side independently of $m$. Therefore one may choose a constant $R_2=R_2(\Vert g{\Vert }_2,\gamma ,\nu ,{\mu }_1)$ such that

Solved by gpt-5.6-sol high.

Use on ${\tilde{H}}_m$ the inner product $(w,z{)}_1=(\nabla w,\nabla z)$. The map $F_m$ in the hint is continuous because it is a finite-dimensional polynomial map. Since the velocity $v={\nabla }^{\perp }\Phi$ is divergence-free,The Poincare inequality bounds $\Vert w{\Vert }_2\le {\mu }_1^{-1/2}\Vert \nabla w{\Vert }_2$. Thus $(F_m(w),w{)}_1<0$ on every $H^1$ sphere whose radius is larger than $\Vert g{\Vert }_2/(\nu \sqrt{{\mu }_1})$. The Brouwer inward-pointing zero lemma supplies $w_{\ast }$ inside that sphere with $F_m(w_{\ast })=0$.

Set ${\omega }_m=w_{\ast }$, solve $-\Delta {\Psi }_m={\omega }_m$ in ${\tilde{H}}_m$, and put $u_m={\nabla }^{\perp }{\Psi }_m$. Expanding $F_m(w_{\ast })=0$ gives exactly the first equation of the Galerkin system, so $({\omega }_m,{\Psi }_m,u_m)$ is a solution.

Solved by gpt-5.6-sol high.