Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence

Colombo, Maria and Crippa, Gianluca and Sorella, Massimo. (2022) Anomalous dissipation and lack of selection in the Obukhov-Corrsin theory of scalar turbulence. Preprints Fachbereich Mathematik, 2022 (08).


Official URL: https://edoc.unibas.ch/89076/

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The Obukhov-Corrsin theory of scalar turbulence [Obu49, Cor51] advances quantitative predictions on passive-scalar advection in a turbulent regime and can be regarded as the analogue for passive scalars of Kolmogorov's K41 theory of fully developed turbulence [Kol41]. The scaling analysis of Obukhov and Corrsin from 1949-1951 identifies a critical regularity threshold for the advection-diffusion equation and predicts anomalous dissipation in the limit of vanishing diffusivity in the supercritical regime. In this paper we provide a fully rigorous mathematical validation of this prediction by constructing a velocity field such that the unique bounded solution of the advection-diffusion equation is bounded uniformly-in-diffusivity in the full supercritical Obukhov-Corrsin regularity regime and exhibits anomalous dissipation. We also show that for a velocity field in $C^{\alpha}$ of space and time (for an arbitrary $0 \leq {\alpha} < 1$) neither vanishing diffusivity nor regularization by convolution provide a selection criterion for bounded solutions of the advection equation.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Crippa, Gianluca
Item Type:Preprint
Publisher:Universität Basel
edoc DOI:
Last Modified:25 Jul 2022 11:47
Deposited On:25 Jul 2022 11:47

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