Lectures on the Onsager Conjecture

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. ”...in three dimensions a mechanism for complete dissipation of all kinetic energy, even without the aid of viscosity, is available.” L. Onsager, 1949 1. Lecture 1: motivation, Onsager criticality. 1.1. Onsager’s original conjecture. The motion of an ideal homogeneous (with constant density 1) incompressible fluid is described by the system of Euler equations given by ∂u ∂t + (u · ∇)u = −∇p, (1) ∇ · u = 0, (2) where u is a divergence-free velocity field, and p is the internal pressure. We assume that the fluid domain Ω here is either periodic or the entire space. It is an easy consequence of the antisymmetry of the nonlinear term in (1) and the incompressibility of the fluid that the law of energy conservation holds for smooth solutions: