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On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa

Abstract

We deal with the vanishing viscosity scheme for the transport/continuity equation $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$ drifted by a divergence-free vector field $\boldsymbol{b}$. Under general Sobolev assumptions on $\boldsymbol{b}$, we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8, 28, 29, 30], and rules out the possibility of anomalous dissipation.

On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

Abstract

We deal with the vanishing viscosity scheme for the transport/continuity equation drifted by a divergence-free vector field . Under general Sobolev assumptions on , we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8, 28, 29, 30], and rules out the possibility of anomalous dissipation.

Paper Structure

This paper contains 10 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Notation
  4. The transport equation. Setup of the vanishing viscosity scheme
  5. The transport equation and regular Lagrangian flows
  6. Setup of the vanishing viscosity scheme
  7. Stochastic flows
  8. A Lagrangian approach to the vanishing viscosity scheme
  9. Rates of convergence
  10. An Eulerian approach to the vanishing viscosity scheme

Key Result

Theorem 2.2

Let $\{ \varphi_i\}_{i\in I}\subset L^1(\mathbb{T}^d)$ be a bounded family. Then the following are equivalent:

Theorems & Definitions (30)

  • Definition 2.1: Equi-integrability
  • Theorem 2.2: Dunford-Pettis, de la Vallée-Poussin
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 3.1: Regular Lagrangian flow
  • ...and 20 more