Shvydkoy, Roman Lectures on the Onsager conjecture. (English) Zbl 1210.76086 Discrete Contin. Dyn. Syst., Ser. S 3, No. 3, 473-496 (2010). Summary: 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. Cited in 1 ReviewCited in 26 Documents MSC: 76F02 Fundamentals of turbulence 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 42B37 Harmonic analysis and PDEs Keywords:Euler equation; Navier-Stokes equation; weak solutions; turbulence; Onsager conjecture; Besov spaces; dyadic models PDFBibTeX XMLCite \textit{R. Shvydkoy}, Discrete Contin. Dyn. Syst., Ser. S 3, No. 3, 473--496 (2010; Zbl 1210.76086) Full Text: DOI