Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / a

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.