A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields

Lucas Huysmans; Ayman Rimah Said

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arXiv · 2402.11642

A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields

Lucas Huysmans·Ayman Rimah Said

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Abstract

We develop a new framework for quantitative estimates of passive scalar transport along Sobolev vector fields in $W^{1,p}$, when $p>1$. Our approach is based on Christ-Journ\'{e} singular integral estimates. We show (i) a new stability estimate which quantifies the dependence of the solution on specific frequencies of the initial data; (ii) a new exponential mixing bound in the full DiPerna-Lions well-posedness class; (iii) propagation of logarithmic Fourier regularity of the solution; (iv) quantitative convergence rates for the vanishing diffusivity and mollification limits; and (v) a logarithmic decay rate for the standard DiPerna-Lions commutator.

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Lucas Huysmans, Ayman Rimah Said. 2024-02-18. A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields. https://arxiv.org/abs/2402.11642

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