Advection Diffusion Equations with Sobolev Velocity Field

In this note we study advection-diffusion equations associated to incompressible-W-1,W-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 enhanced 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.