### Abstract

We prove that a solution to Navier-Stokes equations is inL^{2}(0,∞:H^{2}(Ω)) under the critical assumption thatu∈L^{r,r′}, 3/r+2/r′≤1 withr≥3. A boundaryL^{∞}estimate for the solution is derived if the pressure on the boundary is bounded. Here our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate forL^{∞}-norm ofu. Moreover the solution isC^{1,α}continuous up to boundary if the tangential derivatives of the pressure on the boundary are bounded. Then, from the bootstrap argument a local higher regularity theorem follows, that is, the velocity is as regular as the boundary data of the pressure.

Original language | English |
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Pages (from-to) | 211-247 |

Number of pages | 37 |

Journal | Journal of Differential Equations |

Volume | 149 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1998 Nov 1 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics