## Abstract

We first represent the pressure in terms of the velocity in R_{+}^{3}. Using this representation we prove that a solution to the Navier-Stokes equations is in L^{∞} (R_{+}^{3} × (0, ∞)) under the critical assumption that u ∈ L_{loc}^{r, r′}, frac(3, r) + frac(2, r^{′}) ≤ 1 with r ≥ 3, while for r = 3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L^{∞} estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L^{∞}-norm of u.

Original language | English |
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Pages (from-to) | 2741-2763 |

Number of pages | 23 |

Journal | Journal of Differential Equations |

Volume | 244 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2008 Jun 1 |

### Bibliographical note

Funding Information:Keywords: Slip boundary condition; Navier–Stokes equations; Prodi–Ohyama–Serrin–Ladyzhenskaya condition; Pressure representation; Boundary regularity; Moser iteration * Corresponding author. E-mail addresses: hobae@ajou.ac.kr (H.-O. Bae), choe@yonsei.ac.kr (H.J. Choe), bumjajin@hanmail.net (B.J. Jin). 1 The author is supported by Korean Research Foundation Grant (KRF-2003-015-C00052). 2 The author is supported by the grant (R01-2004-000-100072-0) from the Basic Research Program of the Korea Science & Engineering Foundation.

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics