We consider the transport equation of a passive scalar f(x,t)βR
along a divergence-free vector field u(x,t)βR2, given by
βtβfβ+ββ (uf)=0; and the associated
advection-diffusion equation of f along u, with positive
viscosity/diffusivity parameter Ξ½>0, given by βtβfβ+ββ (uf)βΞ½Ξf=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