### 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)

### Cite this

**On the L _{p}-L_{q} maximal regularity of the Neumann problem for the Stokes equations in a bounded domain.** / Shibata, Yoshihiro; Shimizu, Senjo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

AU - Shibata, Yoshihiro

AU - Shimizu, Senjo

PY - 2008/2/1

Y1 - 2008/2/1

N2 - In this paper, we prove the Lp-Lq 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 Lp-Lq 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.

AB - In this paper, we prove the Lp-Lq 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 Lp-Lq 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.

UR - http://www.scopus.com/inward/record.url?scp=40749116936&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=40749116936&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2008.013

DO - 10.1515/CRELLE.2008.013

M3 - Article

AN - SCOPUS:40749116936

SP - 157

EP - 209

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 615

ER -