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

TL;DR

This paper investigates the vanishing viscosity limit in passive scalar transport equations, showing it fails to select unique or physically admissible solutions, highlighting fundamental issues in the mathematical modeling of such systems.

Contribution

It demonstrates the non-uniqueness and inadmissibility of solutions obtained via vanishing viscosity in passive scalar transport, challenging assumptions about the limit's physical relevance.

Findings

Vanishing viscosity does not select unique solutions.

Vanishing viscosity fails to produce physically admissible solutions.

The limit process can lead to non-physical solution behaviors.

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.