**abstract:**
To try to reach a convenient overview of the state of the art I intend in these three lectures to produce a mixture of both very formal constructions and rigourous theorems. Therefore I will follow the following schedule.
1) Description of the hierarchy of equations from the Hamiltonian systems of Newton Mechanic to models of turbulence, with in between the Boltzmann and Navier Stokes equations. No proof of convergence is given but the classical parameters of fluid mechanic are introduced and the role of entropy is emphasised.
2)The second lecture is devoted to the theorems concerning the incompressible Euler Equation. Local existence with smooth initial data in 3d,
Global existence and uniqueness in 2d with initial bounded vorticity,
Propagation of regularity in 2d with the pair dispersion formula and in 3d with the Beale Kato Majda Kozono criteria and the Constantin Fefferman criteria.
Arnold stability criteria for stationary solutions.
Open problems in particular weak limit of 2d solutions either with oscillating initial data or with vanishing viscosity and no slip boundary condition ( Kato criteria and Grenier instability.)
3) I describe tools used for the macroscopic limit of the Kinetic equations with the hope that they may be used also for other purpose. The averaging lemma of Golse Lions Perthame and Sentis and its L1 extension. The notion of dissipative solution for the incompressible Euler equation application to the incompressible limit of the Boltzmann equation.

Thu 18 Nov, 16:30 - 18:30, Aula Mancini

Fri 19 Nov, 16:30 - 18:30, Aula Mancini

Sat 20 Nov, 16:30 - 18:30, Aula Mancini

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