Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 (Mathematical Analysis of the Incompressible Navier-Stokes Equations)

The sobolev embedding theorem in three dimensions and the periodic elliptic estimate for the Stokes operator giveThe last step uses the absence of the zero Fourier mode: on mean-zero periodic fields, the Poincaré inequality makes the homogeneous $H^2$ norm controlled by $\Vert \Delta v{\Vert }_2=|Av|$.

Because $v$ is a divergence-free vector field, the product rule givesThe integral of a divergence over the periodic domain vanishes. Hence integration by parts yieldsThis is the skew-symmetry of incompressible transport.

Set $\chi =\eta$ in part (ii). The integral is equal to its own negative, soEquivalently, incompressible advection does not change the scalar's quadratic energy.

Both $P_m$ and the Leray-Helmholtz projection are orthogonal projections, hence contractions in $L^2$. Taking the $H$ norm of the first Galerkin equation givesTherefore

The first equation determines $u_m$ linearly from ${\theta }_m$:Substitution into the second equation gives an ordinary differential equation on the finite-dimensional space ${\tilde{H}}_m$. Its right-hand side is polynomial, and therefore locally Lipschitz continuous. The Picard-Lindelöf theorem supplies a unique local solution. The energy estimate in part (iii) bounds ${\theta }_m$ on every finite time interval, so the finite-dimensional continuation criterion rules out finite-time escape. The solution is consequently unique on every interval $[0,T]$.

Take the $L^2$ inner product of the temperature equation with ${\theta }_m$. The spectral projection disappears against ${\theta }_m\in {\tilde{H}}_m$, and part (a)(iii) cancels transport. ThusThe periodic Poincaré inequality, part (i), and the Cauchy-Schwarz inequality implyThe Gronwall inequality therefore gives, for $0\le t\le T$,This defines a bound $K_0(T)$ independent of $m$, and part (i) then givesIntegrating the energy identity and using the same bound on its right-hand side gives

It remains to estimate the time derivative. For $\phi \in H_{\mathrm{per}}^1$, the Fourier projection ${\Pi }_m$ is a contraction in $H^1$, and the skew identity from part (a) givesThe sobolev embedding theorem and the periodic elliptic estimate for the Stokes operator bound $\Vert u_m{\Vert }_{\infty }$ by $C|A{u}_m|$. Moreover,The already obtained bounds therefore implywith $K_0^{\prime }(T)$ independent of $m$.

The uniform bounds and the Banach-Alaoglu theorem provide a subsequence, not relabelled, and a function $\theta$ such thatandSince $H_{\mathrm{per}}^1$ embeds compactly into $L_{\mathrm{per}}^2$, the Aubin-Lions lemma strengthens the first convergence toThe weak continuity from evolution-space bounds gives a representativeTesting against fixed spatial modes and using ${\theta }_m(0)={\Pi }_m{\theta }_0$ shows that this representative satisfies $\theta (0)={\theta }_0$ weakly.

DefineThe first equation and the bounded inverse of the Stokes operator show that $u\in L^{\infty }(0,T;D(A))$ and thatin $L^{\infty }(0,T;H)$. In fact, the strong convergence from part (i) and convergence of the spectral projections implyIn three dimensions $D(A)\subset H^2\hookrightarrow L^{\infty }$, so $u_m{\theta }_m\to u\theta$ in $L^1(0,T;L^2)$. Because $\nabla \cdot u_m=0$,and the nonlinear term consequently converges in distributions and in the required weak $L^2(0,T;H_{\mathrm{per}}^{-1})$ sense. The linear terms pass by weak convergence, while ${\Pi }_m$ tends strongly to the identity. Hencein $L^2(0,T;H_{\mathrm{per}}^{-1})$. Together with part (i), this proves existence of a global weak solution of the Rayleigh-Bénard convection system on every finite interval.

Let $(u_1,{\theta }_1)$ and $(u_2,{\theta }_2)$ be two weak solutions with the same initial data, and set $U=u_1-u_2$ and $\Theta ={\theta }_1-{\theta }_2$. The diagnostic Stokes equation givesSubtracting the temperature equations yieldsPair this equation with $\Theta$. The term transported by $u_1$ vanishes by the skew-symmetry of incompressible transport, while the other nonlinear term satisfiesThe forcing difference is at most $C\Vert \Theta {\Vert }_2^2$. ConsequentlyThe coefficient is integrable on $[0,T]$ because ${\theta }_2\in L^2(0,T;H^1)$. Since $\Theta (0)=0$, the Gronwall inequality gives $\Theta =0$, and the Stokes equation then gives $U=0$. The weak solution is unique.

Part (c)(i) already gives weak continuity of $t\mapsto \theta (t)$ in $L^2$. In the assumed energy equality, the dissipation integralis absolutely continuous. The forcing integrand is in $L^1(0,T)$ because $u\in L^{\infty }(0,T;L^2)$ and $\theta \in L^{\infty }(0,T;L^2)$. The equality therefore makes $t\mapsto \Vert \theta (t){\Vert }_2^2$ continuous.

Whenever $t_n\to t$, weak continuity gives $\theta (t_n)\rightharpoonup \theta (t)$ and the energy equality gives convergence of their norms. The Radon-Riesz theorem, or directly the strong continuity from weak continuity and an energy equality, now gives $\theta (t_n)\to \theta (t)$ in $L^2$. Thus

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

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 Poincaré 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.

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

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:

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

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.