### Abstract

In a domain ω ⊂ R^{n}, consider a weak solution u of the Navier-Stokes equations in the class u ε L∞(0, T;L^{n}(ω)). If lim sup_{t→t*}-0 ||u(t)||^{n}_{n}-||u(t_{*})||^{n}_{n} is small at each point of t_{*} ε (0, T), then u is regular on ω̄ × (0, T). As an application, we give a precise characterization of the singular time; i.e., we show that if a solution u of the Navier-Stokes equations is initially smooth and loses its regularity at some later time T_{*} < T, then either lim sup_{t→T*-0} ||u(t)||L^{n}(ω) = +∞, or u(t) oscillates in L^{n}(ω) around the weak limit wlim_{t→ T*-0} u(t) with sufficiently large amplitude. Furthermore, we prove that every weak solution u of bounded variation on (0, T) with values in L^{n}(ω) becomes regular.

Original language | English |
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Pages (from-to) | 535-554 |

Number of pages | 20 |

Journal | Advances in Differential Equations |

Volume | 2 |

Issue number | 4 |

Publication status | Published - 1997 Dec 1 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Advances in Differential Equations*,

*2*(4), 535-554.