Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 (Mathematical Analysis of the Incompressible Navier-Stokes Equations)

Expand $P_{N}w$ in the orthonormal Stokes operator eigenbasis:The $H$ and $V$ norms satisfySince ${\lambda }_k\le {\lambda }_N$ on the projected space,This is the basic inverse estimate for a spectral projection of the Stokes operator.

Interpolation between $L^2$ and $L^6$, followed by the three-dimensional Sobolev inequality, givesApply this with $v=P_{N}w$ and use part i:Orthogonal projection is contractive in $H$, soThe Sobolev constant is dimensionless after using the periodic-domain normalization, so the estimate is scale invariant.

By the Holder inequality with exponents $3,2,6$,Part ii and the Sobolev embedding $H^1\hookrightarrow L^6$ yieldEquivalently,

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 $m$.

The estimates from part b make $(u_m)$ bounded inand $({\partial }_t{u}_m)$ bounded in $L^2(0,T;V^{\prime })$. Since the periodic embedding $V⋐H$ is compact, the Aubin-Lions lemma supplies a subsequence such thatThe derivatives converge weakly in $L^2(0,T;V^{\prime })$ to ${\partial }_{t}u$.

Because $P_{N}H$ is fixed and finite dimensional, strong convergence in $H$ impliesCombining this with the uniform $L^{\infty }H$ and $L^{2}V$ bounds in the estimate from part a identifies the weak limitPassing to the limit in the Galerkin identity givesThe projected initial data converge to $u_0$ in $H$, so $u(0)=u_0$.

The weak continuity from evolution-space bounds givesSince $T$ was arbitrary, this is a global weak solution of the Navier-Stokes equation with spectrally truncated advection.

The evolution-space regularity permits pairing the equation with $u(t)$ and givesfor almost every $t$. Since $P_{N}u$ is divergence free, periodic integration by parts givesIntegration in time, using $u(0)=u_0$, yields the exact energy balancefor every $t\in [0,T]$. Unlike the usual three-dimensional Leray construction, the improved $L^2(0,T;V^{\prime })$ control of the truncated nonlinearity permits an equality rather than only an inequality.

The energy equality shows thatis continuous. Part i gives weak continuity in $H$. In a Hilbert space, weak convergence together with convergence of norms implies strong convergence. Applying this whenever $t_n\to t$ proves the strong continuity from weak continuity and an energy equality:

Let $u$ and $v$ be two weak solutions with the same initial value, and put $w=u-v$. Bilinearity givesPair with $w$. The first transport term vanishes because $P_{N}u$ is divergence free. Part a and the Young inequality giveThereforeThe coefficient is integrable because $v\in L^2(0,T;V)$. Since $w(0)=0$, the Gronwall inequality gives $w=0$ on $[0,T]$. The global weak solution is unique.

For smooth periodic fields, the Holder inequality and $H^1\hookrightarrow L^6$ giveSimilarly,Adding the bounds proves that the skew-symmetrized transport form extends continuously from smooth fields to ${\tilde{V}}^3$ andThe periodic Poincare inequality givesHence

Periodic integration by parts givesAdding one half of the divergence term to both transport forms yieldsDensity extends the identity from smooth fields to all $u,v,w\in \tilde{V}$. In particular,without requiring $\nabla \cdot u=0$.

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.

The uniform estimate gives a subsequenceThe compact periodic embedding $\tilde{V}⋐\tilde{H}$ improves this toOne may also take strong convergence in $L^3$ by the Rellich-Kondrachov compactness theorem. Combining this with weak convergence of the gradients and using the skew form when derivatives must be moved to a test function givesFor any test function in a fixed finite-dimensional subspace, the Galerkin equation therefore passes to the limit. Density then givesThis is a weak solution of the steady skew-symmetrized Navier-Stokes equation.

Initially $u\in H^1(\Omega )$, so in three dimensionsBoth terms intherefore belong to $L^{3/2}$. The equation becomesPeriodic elliptic regularity givesThe Sobolev embedding then yieldsConsequently each product in $\tilde{B}(u,u)$ belongs to $L^2$, becauseA second application of elliptic regularity now givesEvery term in the equation belongs to $\tilde{H}$, and hence