Weakly Vanishing Divergence-Free Fields with Nonzero Projected Quadratic Flux

A smooth periodic divergence-free sequence converges weakly to zero while its quadratic tensor converges to a nonzero defect tensor. The limiting divergence has nonzero curl and therefore survives the Leray projection. The example concerns weak convergence of velocity fields and does not assert that the fields solve one fixed Navier-Stokes problem.