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

Lucas Huysmans, Edriss S. Titi

TL;DR

This work shows that vanishing viscosity fails as a selection principle for the passive scalar transport equation with bounded divergence-free velocity. By constructing explicit velocity fields and carefully designed vanishing-viscosity sequences, the authors demonstrate both non-uniqueness (multiple inviscid limits from the same data) and inadmissibility (limits that violate entropy and Markovianity). The analysis hinges on renormalised weak solutions, Lagrangian flows, and a framework linking ADE and TE dynamics to control the vanishing-viscosity limit via shear-based constructions. The results imply that diffusion alone cannot guarantee physically meaningful selection of inviscid solutions and highlight the need for alternative selection criteria or regularization approaches. Overall, the paper provides deep insights into the limits of vanishing diffusion in hyperbolic transport and its implications for energy/entropy structure in passive scalar dynamics.

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 $ν>0$, to the PDE $\frac{\partial f}{\partial t} + \nabla\cdot (u f) -νΔf = 0$. 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.

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

TL;DR

This work shows that vanishing viscosity fails as a selection principle for the passive scalar transport equation with bounded divergence-free velocity. By constructing explicit velocity fields and carefully designed vanishing-viscosity sequences, the authors demonstrate both non-uniqueness (multiple inviscid limits from the same data) and inadmissibility (limits that violate entropy and Markovianity). The analysis hinges on renormalised weak solutions, Lagrangian flows, and a framework linking ADE and TE dynamics to control the vanishing-viscosity limit via shear-based constructions. The results imply that diffusion alone cannot guarantee physically meaningful selection of inviscid solutions and highlight the need for alternative selection criteria or regularization approaches. Overall, the paper provides deep insights into the limits of vanishing diffusion in hyperbolic transport and its implications for energy/entropy structure in passive scalar dynamics.

Abstract

We study selection by vanishing viscosity for the transport of a passive scalar advected by a bounded, divergence-free vector field . This is described by the initial value problem to the PDE , or with positive viscosity/diffusivity , to the PDE . 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.
Paper Structure (16 sections, 15 theorems, 207 equations)

This paper contains 16 sections, 15 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Entropy of inviscid transport
  5. Previous work on anomalous dissipation
  6. Previous work on non-uniqueness
  7. Further comments
  8. Outline of paper
  9. Notation and definitions
  10. Notation
  11. Necessary background
  12. Vanishing viscosity
  13. Non-uniqueness
  14. Notation
  15. Inadmissibility
  16. ...and 1 more sections

Key Result

Theorem \ref{vvtheorem}

There exists a divergence-free vector field $u \in L^\infty([0,1];L^\infty(\mathbb{T}^2;\mathbb{R}^2))$, and a sequence $\{\nu_n\}_{n\in\mathbb{N}}$ with $\nu_n > 0$ and $\nu_n\xrightarrow{n\to\infty}0$, such that for any initial data $f_0\in L^\infty(\mathbb{T}^2)$, and for $f^\nu$ the unique solu with the above convergence in weak-$*$$L^\infty([0,1];L^\infty(\mathbb{T}^2))$, and strong in $L^p(

Theorems & Definitions (50)

  • Theorem \ref{vvtheorem}: Non-unique renormalised vanishing viscosity limit
  • Theorem \ref{perfmixvv}: Inadmissible vanishing viscosity limit
  • Definition \ref{perfmixvv}: Weak solution to the transport equation \ref{['eqTE']}
  • Remark \ref{perfmixvv}
  • Definition \ref{perfmixvv}: Renormalised weak solutions to \ref{['eqTE']}
  • Remark \ref{perfmixvv}
  • Definition \ref{perfmixvv}: Lagrangian flows and Lagrangian solutions to \ref{['eqTE']}
  • Remark \ref{perfmixvv}
  • Definition \ref{perfmixvv}: Weak solution to the advection-diffusion equation \ref{['eqADE']}
  • Definition \ref{perfmixvv}: Vanishing viscosity solution to \ref{['eqTE']}
  • ...and 40 more