Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / c

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