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These lectures give an account of recent results pertaining to the celebrated Onsager conjecture. The conjecture states that the minimal space regularity needed for a weak solution of the Euler equation to conserve energy is $1/3$. Our presentation is based on the Littlewood-Paley method. We start with quasi-local estimates on the energy flux, introduce Onsager criticality, find a positive solution to the conjecture in Besov spaces of smoothness $1/3$. We illuminate important connections with the scaling laws of turbulence. Results for dyadic models and a complete resolution of the Onsager conjecture for those is discussed, as well as recent attempts to construct dissipative solutions for the actual equation.
The article is based on a series of four lectures given at the 11th school "Mathematical Theory in Fluid Mechanics" in Kácov, Czech Republic, May 2009.