Abstract
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.
Highlights
These lectures give an account of recent results pertaining to the celebrated Onsager conjecture
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: (3)
Π, and any better regularity would be sufficient to justify integration by parts in (4) to show that Π = 0. It is exactly what Onsager conjectured in his seminal paper on statistical fluid dynamics [42]: a) every weak solution to the Euler equations with smoothness h > 1/3 does not dissipate energy; b) and there exists a weak solution to (1) - (2) of smoothness of exactly 1/3 which does not conserve energy
Summary
Π, and any better regularity would be sufficient to justify integration by parts in (4) to show that Π = 0 It is exactly what Onsager conjectured in his seminal paper on statistical fluid dynamics [42]: a) every weak solution to the Euler equations with smoothness h > 1/3 does not dissipate energy; b) and there exists a weak solution to (1) - (2) of smoothness of exactly 1/3 which does not conserve energy. Since in the Euler equation (1) the nonlinear term is the only term present in the energy budget we again are led to Onsager’s claim Due to this special relevance to turbulence it makes sense to state the Onsager conjecture more broadly for a forced equation. We would like to find a bound on it in terms of u
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