Non-uniqueness & inadmissibility of the vanishing viscosity limit of the passive scalar transport equation

We study the vanishing viscosity/diffusivity limit for the transport of a passive scalar $f(x,t)\in R$ by a bounded, divergence-free vector field $u(x,t)\in R^2$. This is described by the Cauchy problem to the PDE $\frac{\partial f}{\partial t}+\nabla \cdot \left(uf\right)=0$, or with viscosity $\nu >0$, to the PDE $\frac{\partial f}{\partial t}+\nabla \cdot \left(uf\right)-\nu \Delta f=0$. In the first part of this work, we construct a bounded, divergence-free vector field $u(x,t)$ for which, for any non-constant initial datum, the viscous solutions along different subsequences of the vanishing viscosity limit converge to different solutions to the inviscid problem. In the second part, we construct another bounded, divergence-free vector field $u(x,t)$ for which, for every initial datum, the vanishing viscosity limit of solutions exists, is unique, and converges to an inviscid solution; however, when the initial datum is not constant, this inviscid limit is physically inadmissible due to increasing energy/entropy.