Lucas Huysmans (Cambridge): Non-uniqueness and inadmissibility of the vanishing diffusion limit for passive scalar transport

We study the vanishing diffusivity limit for the passive scalar transport equation along a bounded divergence-free vector field on the two-dimensional torus. We construct two velocity vector fields each exhibiting peculiar behaviour of the vanishing diffusion limit. For the first we demonstrate that along different subsequences of diffusivities the limit converge strongly to different solutions to inviscid transport for any (non-constant) initial data. Both of these limits are renormalised solutions to the transport equation, and so equally physically admissible. For the second vector field we show uniqueness of the vanishing diffusivity limit, however, for any initial data, the unique limit perfectly mixes to its spatial average and after a short delay perfectly unmixes to its original state. Therefore the limit exhibits a dissipation of energy/entropy but later a reverse of that dissipation.