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

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

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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.

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The second estimate in part (a) bounds $({\omega }_m)$ in $H_{\mathrm{per}}^2$. The periodic Poisson equation $-\Delta {\Psi }_m={\omega }_m$ and the supplied curl identity then bound $({\Psi }_m)$ in $H_{\mathrm{per}}^4$ and $(u_m)$ in $H_{\mathrm{per}}^3$. After passing to a subsequence,The Rellich-Kondrachov compactness theorem also gives strong convergence in the corresponding spaces with one fewer derivative. In particular, $u_m\to u$ in $L^{\infty }$ and $\nabla {\omega }_m\to \nabla \omega$ in $L^2$, soPassing to the limit in the Galerkin equations givesThese identities have the claimed Sobolev regularity, and the first holds in $L_{\mathrm{per}}^2$. Finally, the weak lower semicontinuity of the Hilbert norm preserves the estimates

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The regularity from part (i) makes $u\omega$ and its first weak derivatives square-integrable. The product rule therefore holds in the distributional derivative sense:Since $u={\nabla }^{\perp }\Psi$, equality of mixed weak derivatives gives $\nabla \cdot u=0$. Both remaining expressions belong to $L_{\mathrm{per}}^2$, so their distributional equality is an equality in that space:

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Let $q=2p+2$ and multiply the vorticity equation by ${\omega }^{q-1}$, first for an odd power as in the hint. Periodic integration by parts and $\nabla \cdot u=0$ giveThe diffusion term is nonnegative, while the Holder inequality bounds the right-hand side by $\Vert g{\Vert }_q\Vert \omega {\Vert }_q^{q-1}$. ConsequentlyOn the finite-volume torus, $L^q$ norms increase to the essential supremum as $q\to \infty$. Hence the damped-vorticity maximum estimate gives

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For each $\nu >0$, choose the solution from parts (a)--(c). The maximum estimate gives a $\nu$-independent boundThus a sequence ${\nu }_j\downarrow 0$ has ${\omega }_{{\nu }_j}{\rightharpoonup }^{\ast }\omega$ in $L^{\infty }$. The periodic elliptic estimates forbound ${\Psi }_{{\nu }_j}$ in $W^{2,p}$ and $u_{{\nu }_j}$ in $W^{1,p}$ for every finite $p$. Taking $p>2$ and using compact sobolev embedding gives, after a further subsequence, uniform convergence $u_{{\nu }_j}\to u$. It follows that $u_{{\nu }_j}{\omega }_{{\nu }_j}\rightharpoonup u\omega$ distributionally.

The first energy estimate givesTherefore the viscous term vanishes in $H^{-1}$, and the weak formulation passes to the limit asThe elliptic relations pass to the limit as well and giveSince $\omega \in L^{\infty }\subset L^2$, periodic elliptic regularity gives $\Psi \in H_{\mathrm{per}}^2$ and $u\in H_{\mathrm{per}}^1\cap H$. Moreover $u\omega \in L^2$, so its divergence lies in $H_{\mathrm{per}}^{-1}$. This constructs the required weak solution of the damped-driven Euler system by a vanishing-viscosity limit.

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