30 research outputs found
Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation
We consider the transport equation of a passive scalar
along a divergence-free vector field , given by
; and the associated
advection-diffusion equation of along , with positive
viscosity/diffusivity parameter , given by . We demonstrate failure of the
vanishing viscosity limit of advection-diffusion to select unique solutions, or
to select entropy-admissible solutions, to transport along .
First, we construct a bounded divergence-free vector field 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
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
NON-UNIQUENESS & INADMISSIBILITY OF THE VANISHING VISCOSITY LIMIT OF THE PASSIVE SCALAR TRANSPORT EQUATION
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
Barbarians at the British Museum: Anglo-Saxon Art, Race and Religion
A critical historiographical overview of art historical approaches to early medieval material culture, with a focus on the British Museum collections and their connections to religion