## Abstract

We consider the Cauchy problem for the Navier–Stokes equation in ℝ^{3}×]0,∞[ with the initial datum u_{0} ∈ L^{3} _{weak}, a critical space containing nontrivial (−1)−homogeneous fields. For small ||u_{0}||L^{3} _{weak} one can get global well-posedness by perturbation theory. When ||u_{0}||L^{3} _{weak} is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u^{(n)} are weak solutions corresponding the the initial datum u^{(n)} _{0}, and u^{(n)} _{0} converge weakly* in L^{3} _{weak} to u_{0}, then a suitable subsequence of u^{(n)} converges to a weak solution u corresponding to the initial condition u_{0}. This is of interest even in the special case u_{0}≡0.

Original language | English (US) |
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Pages (from-to) | 628-651 |

Number of pages | 24 |

Journal | Communications in Partial Differential Equations |

Volume | 43 |

Issue number | 4 |

DOIs | |

State | Published - Apr 3 2018 |

### Bibliographical note

Publisher Copyright:© 2018, © 2018 Taylor & Francis.

## Keywords

- Large initial data in critical spaces
- Navier–Stokes solutions
- stability of weak solutions

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