# Advection diffusion equations with Sobolev velocity field

created by bruè on 18 Mar 2020
modified by nguyen on 21 Feb 2021

[BibTeX]

Accepted Paper

Inserted: 18 mar 2020
Last Updated: 21 feb 2021

Journal: Communications in Mathematical Physics
Year: 2020

Abstract:

In this note we study advection diffusion equations associated to incompressible $W^{1,p}$ velocity fields with $p>2$. We present new estimates on the energy dissipation rate and we discuss applications to the study of upper bounds on the enhancing dissipation rate, lower bounds on the $L^2$ norm of the density, and quantitative vanishing viscosity estimates. The key tools employed in our argument are a propagation of regularity result, coming from the study of transport equations, and a new result connecting the energy dissipation rate to regularity estimates for transport equations. Eventually we provide examples which underline the sharpness of our estimates.