### Abstract

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus T^{d}, assuming that the solutions have norms for Besov space B^{σ,&inf;} _{3} (T^{d}), σ &insin; (0, 1], that are bounded in the L^{3}-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(v(3σ-1)/(σ+1)), vanishing as v ← 0 if σ > 1/3. A consequence is that Onsager-type 'quasi-singularities' are required in the Leray solutions, even if the total energy dissipation vanishes in the limit v ← 0, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For σ &insin; (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially 'rough' but conserve energy.

Original language | English (US) |
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Pages (from-to) | 4465-4482 |

Number of pages | 18 |

Journal | Nonlinearity |

Volume | 32 |

Issue number | 11 |

DOIs | |

State | Published - Oct 10 2019 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Keywords

- anomalous dissipation
- fluid turbulence
- Onsager's conjecture

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## Cite this

*Nonlinearity*,

*32*(11), 4465-4482. https://doi.org/10.1088/1361-6544/ab2f42