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September  2010, 3(3): 473-496. doi: 10.3934/dcdss.2010.3.473

Lectures on the Onsager conjecture

Roman Shvydkoy 1,

1. 

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St. M/C 249 Chicago, IL 60607-7045, United States

Received  October 2009 Revised  January 2010 Published  May 2010

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.
Keywords: Navier-Stokes equation, Onsager conjecture, Euler equation, dyadic models., weak solutions, Besov spaces, turbulence.
Mathematics Subject Classification: Primary: 76F02, 76B03; Secondary: 42B3.
Citation: Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 473-496. doi: 10.3934/dcdss.2010.3.473
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