### Abstract

Consider the Navier-Stokes equations in Ω × (0,T), where Ω is a domain in R^{3}. We show that there is an absolute constant ε_{0} such that every weak solution u with the property that sup_{t∈(a,b)} ∥u(t)∥_{L3W(D)} ≤ ε_{0} is necessarily of class C^{∞} in the space-time variables on any compact subset of D x (a, b), where D ⊂⊂ Ω and 0 < a < b < T. As an application, we prove that if the weak solution u behaves around (x_{0}, t_{0}) ∈ Ω × (0,T) like u(x,t) = 0(|x - x_{0}l^{-1}) as x → x_{0} uniformly in t in some neighbourhood of t_{0}, then (x_{0},t_{0}) is actually a removable singularity of u.

Original language | English |
---|---|

Pages (from-to) | 949-966 |

Number of pages | 18 |

Journal | Communications in Partial Differential Equations |

Volume | 23 |

Issue number | 5-6 |

Publication status | Published - 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**Removable singularities of weak solutions to the Navier-Stokes equations.** / Kozono, Hideo.

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 23, no. 5-6, pp. 949-966.

}

TY - JOUR

T1 - Removable singularities of weak solutions to the Navier-Stokes equations

AU - Kozono, Hideo

PY - 1998

Y1 - 1998

N2 - Consider the Navier-Stokes equations in Ω × (0,T), where Ω is a domain in R3. We show that there is an absolute constant ε0 such that every weak solution u with the property that supt∈(a,b) ∥u(t)∥L3W(D) ≤ ε0 is necessarily of class C∞ in the space-time variables on any compact subset of D x (a, b), where D ⊂⊂ Ω and 0 < a < b < T. As an application, we prove that if the weak solution u behaves around (x0, t0) ∈ Ω × (0,T) like u(x,t) = 0(|x - x0l-1) as x → x0 uniformly in t in some neighbourhood of t0, then (x0,t0) is actually a removable singularity of u.

AB - Consider the Navier-Stokes equations in Ω × (0,T), where Ω is a domain in R3. We show that there is an absolute constant ε0 such that every weak solution u with the property that supt∈(a,b) ∥u(t)∥L3W(D) ≤ ε0 is necessarily of class C∞ in the space-time variables on any compact subset of D x (a, b), where D ⊂⊂ Ω and 0 < a < b < T. As an application, we prove that if the weak solution u behaves around (x0, t0) ∈ Ω × (0,T) like u(x,t) = 0(|x - x0l-1) as x → x0 uniformly in t in some neighbourhood of t0, then (x0,t0) is actually a removable singularity of u.

UR - http://www.scopus.com/inward/record.url?scp=0038815943&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038815943&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 949

EP - 966

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 5-6

ER -