Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation

We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in \mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in {\mathbb{R}}^2$. This is described by the initial value problem to the PDE $\frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (uf)=0$, or with positive viscosity/diffusivity $\nu >0$, to the PDE $\frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (uf)-\nu \mathrm{\Delta }f=0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy.