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

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 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

*Journal of Differential Equations*,

*244*(11), 2741-2763. https://doi.org/10.1016/j.jde.2008.02.039

}

*Journal of Differential Equations*, vol. 244, no. 11, pp. 2741-2763. https://doi.org/10.1016/j.jde.2008.02.039

**Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition.** / Bae, Hyeong Ohk; Choe, Hi Jun; Jin, Bum Ja.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

AU - Bae, Hyeong Ohk

AU - Choe, Hi Jun

AU - Jin, Bum Ja

PY - 2008/6/1

Y1 - 2008/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=41949110368&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41949110368&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.02.039

DO - 10.1016/j.jde.2008.02.039

M3 - Article

AN - SCOPUS:41949110368

VL - 244

SP - 2741

EP - 2763

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 11

ER -