Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible C∞([0,T)×Td)∩L1([0,T];C1-(Td))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))$$\end{document} velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible L1([0,T];C1-(Td))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1([0,T]; C^{1-}({\mathbb {T}}^d))$$\end{document} drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.