## Abstract

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in ℝ^{3} conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space B_{3,c(ℕ)} ^{1/3}. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to B_{3,c(ℕ)}^{1/3} conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Original language | English (US) |
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Pages (from-to) | 1233-1252 |

Number of pages | 20 |

Journal | Nonlinearity |

Volume | 21 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2008 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics