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 ${\partial }_{t}u+\text{div }(u\mathbf{b})=0$ drifted by a divergence-free vector field $\mathbf{b}$. Under general Sobolev assumptions on $\mathbf{b}$, 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.

arXiv comment18 pages. Compared to the previous version, section 3 has been moved to a different paper, while section 4 has been moved to the appendix of the present work

References in corpus (8)

• Lagrangian flows for vector fields with gradient given by a singular integral
• Convex integration solutions to the transport equation with full dimensional concentration
• Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit
• Advection diffusion equations with Sobolev velocity field
• Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy
• Smooth approximation is not a selection principle for the transport equation with rough vector field
• Quantitative stability estimates for Fokker-Planck equations
• Anomalous Dissipation in Passive Scalar Transport

Cited by in corpus (4)

• LDP and CLT for SPDEs with Transport Noise
• Weak and parabolic solutions of advection-diffusion equations with rough velocity field
• A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion
• Propagation of logarithmic regularity and inviscid limit for the 2D Euler equations