### Abstract

In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter λεΣ∈,γ0, where Σ∈,γ0={λεC||argλ|≤π-∈,|λ|≥γ0}(0<∈<πâ•2,γ0>0), and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ∈ < π / 2 and γ<inf>0</inf> > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ<inf>0</inf> > 0 for given 0 < ∈ < π / 2. We also prove the maximal L<inf>p</inf> - L<inf>q</inf> regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x′=(x1,...,xN-1).

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

Pages (from-to) | 1888-1925 |

Number of pages | 38 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 38 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2015 Jun 1 |

### Fingerprint

### Keywords

- infinite layer
- maximal regularity
- R -boundedness
- Stokes equations

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

**On the R -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer.** / Saito, Hirokazu.

Research output: Contribution to journal › Article

*Mathematical Methods in the Applied Sciences*, vol. 38, no. 9, pp. 1888-1925. https://doi.org/10.1002/mma.3201

}

TY - JOUR

T1 - On the R -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

AU - Saito, Hirokazu

PY - 2015/6/1

Y1 - 2015/6/1

N2 - In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter λεΣ∈,γ0, where Σ∈,γ0={λεC||argλ|≤π-∈,|λ|≥γ0}(0<∈<πâ•2,γ0>0), and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ∈ < π / 2 and γ0 > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ0 > 0 for given 0 < ∈ < π / 2. We also prove the maximal Lp - Lq regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x′=(x1,...,xN-1).

AB - In this paper, we prove the R-boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter λεΣ∈,γ0, where Σ∈,γ0={λεC||argλ|≤π-∈,|λ|≥γ0}(0<∈<πâ•2,γ0>0), and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ∈ < π / 2 and γ0 > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ0 > 0 for given 0 < ∈ < π / 2. We also prove the maximal Lp - Lq regularity theorem of the nonstationary Stokes problem as an application of the R-boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable x′=(x1,...,xN-1).

KW - infinite layer

KW - maximal regularity

KW - R -boundedness

KW - Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=84929501870&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929501870&partnerID=8YFLogxK

U2 - 10.1002/mma.3201

DO - 10.1002/mma.3201

M3 - Article

AN - SCOPUS:84929501870

VL - 38

SP - 1888

EP - 1925

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 9

ER -