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

We consider the transport equation of a passive scalar $f(x,t)\in\mathbb{R}$ along a divergence-free vector field $u(x,t)\in\mathbb{R}^2$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$; and the associated advection-diffusion equation of $f$ along $u$, with positive viscosity/diffusivity parameter $\nu>0$, given by $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -\nu\Delta f = 0$. We demonstrate failure of the vanishing viscosity limit of advection-diffusion to select unique solutions, or to select entropy-admissible solutions, to transport along $u$. First, we construct a bounded divergence-free vector field $u$ which has, for each (non-constant) initial datum, two weak solutions to the transport equation. Moreover, we show that both these solutions are renormalised weak solutions, and are obtained as strong limits of a subsequence of the vanishing viscosity limit of the corresponding advection-diffusion equation. Second, we construct a second bounded divergence-free vector field $u$ admitting, for any initial datum, a weak solution to the transport equation which is perfectly mixed to its spatial average, and after a delay, unmixes to its initial state. Moreover, we show that this entropy-inadmissible unmixing is the unique weak vanishing viscosity limit of the corresponding advection-diffusion equation.