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The aim of this short course is to provide some general analytical tools to deal with flows of rough vector fields, objects that naturally arise in the study of “turbulent fluids”. We will start discussing the easier case of the linear transport equation, presenting some aspects of the Diperna-Lions and Ambrosio theories. We then pass to the incompressible Euler equations by focusing on its pure and idealized aspects in connection to the Kolmogorov Theory of Turbulence. We discuss the Onsager’s conjecture and conclude describing the main current research directions.
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