### Abstract

In this paper, we prove the L_{p}-L_{q} maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvest. 31 (1988), 381405., but he assumed that p = q > 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 < p, q < but also the infinite time interval case. Especially, we obtain the L_{p}-L_{q} maximal regularity theorem with exponential stability on the infinite time interval. Our method can be applied to any initial boundary value problem for the equation of parabolic type with suitable boundary condition which generates an analytic semigroup, for example the Stokes equation with non-slip, slip or Robin boundary conditions.

Original language | English |
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Pages (from-to) | 157-209 |

Number of pages | 53 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 615 |

DOIs | |

Publication status | Published - 2008 Feb 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)