### Abstract

We show that the time decay rate of L^{2} norm of weak solution for the Stokes equations and for the Navier-Stokes equations on the half spaces are t^{-n/2 (1/r - 1/2) - 1/2} if the initial data u_{0} ∈ L^{2}∩L^{r} and ∫_{ℝn+}\y_{n}u_{0}(y)|^{r} dy < ∞ for 1 < r < 2. We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was t^{-n/2 (1/r - 1/2)}, which was obtained by Borchers and Miyakawa [1] and Ukai [9].

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

Pages (from-to) | 799-816 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 238 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2001 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*238*(4), 799-816. https://doi.org/10.1007/s002090100276

}

*Mathematische Zeitschrift*, vol. 238, no. 4, pp. 799-816. https://doi.org/10.1007/s002090100276

**Decay rate for the incompressible flows in half spaces.** / Bae, Hyeong Ohk; Choe, Hi Jun.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Decay rate for the incompressible flows in half spaces

AU - Bae, Hyeong Ohk

AU - Choe, Hi Jun

PY - 2001/1/1

Y1 - 2001/1/1

N2 - We show that the time decay rate of L2 norm of weak solution for the Stokes equations and for the Navier-Stokes equations on the half spaces are t-n/2 (1/r - 1/2) - 1/2 if the initial data u0 ∈ L2∩Lr and ∫ℝn+\ynu0(y)|r dy < ∞ for 1 < r < 2. We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was t-n/2 (1/r - 1/2), which was obtained by Borchers and Miyakawa [1] and Ukai [9].

AB - We show that the time decay rate of L2 norm of weak solution for the Stokes equations and for the Navier-Stokes equations on the half spaces are t-n/2 (1/r - 1/2) - 1/2 if the initial data u0 ∈ L2∩Lr and ∫ℝn+\ynu0(y)|r dy < ∞ for 1 < r < 2. We also show that the decay rate is determined by the linear part of the weak solution. We use the heat kernel and Ukai's solution formula for the Stokes equations. It has been known up to now that the decay rate on the half space was t-n/2 (1/r - 1/2), which was obtained by Borchers and Miyakawa [1] and Ukai [9].

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U2 - 10.1007/s002090100276

DO - 10.1007/s002090100276

M3 - Article

AN - SCOPUS:0035732263

VL - 238

SP - 799

EP - 816

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 4

ER -