Advection diffusion equations with Sobolev velocity field

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.

References in corpus (5)

• Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows
• Convex integration solutions to the transport equation with full dimensional concentration
• Dissipation Enhancement by Mixing
• Diffusion-limited mixing by incompressible flows
• A quantitative theory for the continuity equation

Cited by in corpus (6)

• Anomalous dissipation for the forced 3D Navier-Stokes equations
• On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity
• Weak and parabolic solutions of advection-diffusion equations with rough velocity field
• Growth of Sobolev norms and loss of regularity in transport equations
• 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