### 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 |
---|---|

Pages (from-to) | 535-554 |

Number of pages | 20 |

Journal | Advances in Differential Equations |

Volume | 2 |

Issue number | 4 |

Publication status | Published - 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Advances in Differential Equations*,

*2*(4), 535-554.

**Regularity criterion on weak solutions to the navier-stokes equations.** / Kozono, Hideo; Sohr, Hermann.

Research output: Contribution to journal › Article

*Advances in Differential Equations*, vol. 2, no. 4, pp. 535-554.

}

TY - JOUR

T1 - Regularity criterion on weak solutions to the navier-stokes equations

AU - Kozono, Hideo

AU - Sohr, Hermann

PY - 1997

Y1 - 1997

N2 - In a domain ω ⊂ Rn, consider a weak solution u of the Navier-Stokes equations in the class u ε L∞(0, T;Ln(ω)). If lim supt→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 supt→T*-0 ||u(t)||Ln(ω) = +∞, or u(t) oscillates in Ln(ω) around the weak limit wlimt→ 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 Ln(ω) becomes regular.

AB - In a domain ω ⊂ Rn, consider a weak solution u of the Navier-Stokes equations in the class u ε L∞(0, T;Ln(ω)). If lim supt→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 supt→T*-0 ||u(t)||Ln(ω) = +∞, or u(t) oscillates in Ln(ω) around the weak limit wlimt→ 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 Ln(ω) becomes regular.

UR - http://www.scopus.com/inward/record.url?scp=0011968025&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011968025&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011968025

VL - 2

SP - 535

EP - 554

JO - Advances in Differential Equations

JF - Advances in Differential Equations

SN - 1079-9389

IS - 4

ER -