NON-UNIQUENESS & INADMISSIBILITY OF THE VANISHING VISCOSITY LIMIT OF THE PASSIVE SCALAR TRANSPORT EQUATION

Abstract

We consider the transport equation of a passive scalar f(x, t) ∈ R along a divergence-free vector field u(x, t) ∈ R2, given by @f @t + ∇ · (uf) = 0; and the associated advection-diffusion equation of f along u, with positive viscosity/diffusivity parameter � > 0, given by @f @t + ∇ · (uf) − ��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.