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arXiv · 2003.08198

Advection diffusion equations with Sobolev velocity field

Elia Bruè·Quoc-Hung Nguyen
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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.

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Elia Bruè, Quoc-Hung Nguyen. 2020-03-18. Advection diffusion equations with Sobolev velocity field. https://doi.org/10.1007/s00220-021-03993-4

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