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

TL;DR

This paper investigates the vanishing viscosity method for the transport equation with rough divergence-free coefficients, proving convergence to the unique Lagrangian solution and providing quantitative convergence rates.

Contribution

It establishes the convergence of the vanishing viscosity scheme under general Sobolev conditions and introduces a new selection criterion for solutions beyond the distributional regime.

Findings

Proves convergence of the scheme to the unique Lagrangian solution.

Provides quantitative rates of convergence.

Rules out anomalous dissipation in the considered setting.

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.