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Non-Uniqueness and Inadmissibility of the Vanishing Viscosity Limit of the Passive Scalar Transport Equation
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abstract
We study selection by vanishing viscosity for the transport of a passive scalar $f(x,t)\in\mathbb{R}$ advected by a bounded, divergence-free vector field $u(x,t)\in\mathbb{R}^2$. This is described by the initial value problem to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) = 0$, or with positive viscosity/diffusivity $\nu>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -\nu\Delta f = 0$. We demonstrate the failure of the vanishing viscosity limit to select (a) unique solutions or (b) physically admissible solutions in the sense of non-increasing energy/entropy.
Forward citations
Cited by 2 Pith papers
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Spontaneous stochasticity and the Armstrong-Vicol passive scalar
Eulerian spontaneous stochasticity is defined as liminf less than limsup of observables in the inviscid limit; all probability measures on the attainable set are shown selectable, and the Armstrong-Vicol passive scala...
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On vanishing diffusivity selection for the advection equation
Vanishing diffusivity uniquely selects the solution of the advection equation for divergence-free BV vector fields singular only at the initial time, including Depauw's non-uniqueness example.
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