Advection Diffusion Equations with Sobolev Velocity Field


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 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.

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  1. 1.

    Here and in the sequel we use the notation \(A \sim _c B\) to mean that \(C^{-1}A \le B \le C A\) where C depends only on c. Similar notation will be adopted for \(\lesssim _c\) and \( > rsim _c\).


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Most of this work was developed while the first author was a PhD student at Scuola Normale Superiore, Pisa. The second author was supported by the ShanghaiTech University startup fund, and part of this work was done while he was visiting Scuola Normale Superiore. The authors wish to express their gratitude to this institution for the excellent working conditions and the stimulating atmosphere.

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Correspondence to Quoc-Hung Nguyen.

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Brué, E., Nguyen, QH. Advection Diffusion Equations with Sobolev Velocity Field. Commun. Math. Phys. (2021).

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