### Abstract

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Original language | English (US) |
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Pages (from-to) | 101-144 |

Number of pages | 44 |

Journal | Annals of Mathematics |

Volume | 189 |

Issue number | 1 |

DOIs | |

State | Published - 2019 |

### Keywords

- Convex integration
- Euler equations
- Intermittency
- Inviscid limit
- Navier-Stokes
- Turbulence
- Weak solutions

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Buckmaster, T., & Vicol, V. (2019). Nonuniqueness of weak solutions to the Navier-Stokes equation.

*Annals of Mathematics*,*189*(1), 101-144. https://doi.org/10.4007/annals.2019.189.1.3