### Abstract

In this paper, we investigate L^{p}-estimates for the solution of the Stokes equation in a half space H where 1 <p <∞. It is shown that the solution of the Stokes equation is governed by an analytic semigroup on B U C_{σ}(H), C_{0,σ}(H) or L_{σ}
^{∞}(H). From the operatortheoretical point of view it is a surprising fact that the corresponding result for L_{σ}
^{1}(H) does not hold true. In fact, there exists an L^{1} -function f satisfying div f = 0 such that the solution of the corresponding resolvent equation with right hand side f does not belong to L^{1} . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the L^{1} -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded H^{∞}-calculus on L^{p} for 1 <p <∞ and obtain as a consequence maximal L^{p}-L^{q}-regularity for the solution of the Stokes equation.

Original language | English |
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Pages (from-to) | 115-142 |

Number of pages | 28 |

Journal | Journal of Evolution Equations |

Volume | 1 |

Issue number | 1 |

Publication status | Published - 2001 |

Externally published | Yes |

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### Keywords

- Analytic semigroups
- H-calculus
- Maximal L-regularity
- R-boundedness
- Resolvent estimates
- Stokes system

### ASJC Scopus subject areas

- Ecology, Evolution, Behavior and Systematics

### Cite this

*Journal of Evolution Equations*,

*1*(1), 115-142.