### Abstract

In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W^{3−1/r} _{r} domain of N-dimensional Euclidean space ℝ^{N} (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σ_{ε,λ0} = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ_{0}} (0 < ε < π/2), and the maximal L_{p}-L_{q} regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W^{1} _{q} (Ω) to the variational problem: (∇p,∇ϕ)_{Ω} = (f,∇ϕ)_{Ω} for any ϕ ∈ W^{1} _{q′}(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W^{1} _{q} (Ω) is a closed subspace of Ŵ^{1} _{q,Γ} (Ω) = {p ∈ L_{q,loc}(Ω) | ∇p ∈ L_{q}(Ω)^{N}, p|_{Γ} = 0} with respect to gradient norm ∥∇ · ∥_{Lq(} _{Ω)} that contains a space W^{1} _{q,} _{Γ} (Ω) = {p ∈ W^{1} _{q} (Ω) | p|_{Γ} = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ_{0},∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W^{1} _{q} (Ω) = Ŵ^{1} _{q,Γ} (Ω).

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

Title of host publication | Mathematical Fluid Dynamics, Present and Future |

Publisher | Springer New York LLC |

Pages | 203-285 |

Number of pages | 83 |

Volume | 183 |

ISBN (Print) | 9784431564553 |

DOIs | |

Publication status | Published - 2016 |

Event | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan Duration: 2014 Nov 11 → 2014 Nov 14 |

### Other

Other | 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 |
---|---|

Country | Japan |

City | Tokyo |

Period | 14/11/11 → 14/11/14 |

### Fingerprint

### Keywords

- Analytic semigroup
- Free boundary condition
- Maximal L-L regularity
- R-Boundedness
- Stokes equations
- Surface tension
- UniformW domain

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{p}-l

_{q}regularity of the stokes equations with free boundary condition. In

*Mathematical Fluid Dynamics, Present and Future*(Vol. 183, pp. 203-285). Springer New York LLC. https://doi.org/10.1007/978-4-431-56457-7_9

**On the r-bounded solution operator and the maximal l _{p}-l_{q} regularity of the stokes equations with free boundary condition.** / Shibata, Yoshihiro.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{p}-l

_{q}regularity of the stokes equations with free boundary condition. in

*Mathematical Fluid Dynamics, Present and Future.*vol. 183, Springer New York LLC, pp. 203-285, 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014, Tokyo, Japan, 14/11/11. https://doi.org/10.1007/978-4-431-56457-7_9

_{p}-l

_{q}regularity of the stokes equations with free boundary condition. In Mathematical Fluid Dynamics, Present and Future. Vol. 183. Springer New York LLC. 2016. p. 203-285 https://doi.org/10.1007/978-4-431-56457-7_9

}

TY - GEN

T1 - On the r-bounded solution operator and the maximal lp-lq regularity of the stokes equations with free boundary condition

AU - Shibata, Yoshihiro

PY - 2016

Y1 - 2016

N2 - In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W3−1/r r domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 < ε < π/2), and the maximal Lp-Lq regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W1 q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1 q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1 q (Ω) is a closed subspace of Ŵ1 q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq( Ω) that contains a space W1 q, Γ (Ω) = {p ∈ W1 q (Ω) | p|Γ = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ0,∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W1 q (Ω) = Ŵ1 q,Γ (Ω).

AB - In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W3−1/r r domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N < r < ∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 < ε < π/2), and the maximal Lp-Lq regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W1 q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1 q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1 q (Ω) is a closed subspace of Ŵ1 q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq( Ω) that contains a space W1 q, Γ (Ω) = {p ∈ W1 q (Ω) | p|Γ = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ0,∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W1 q (Ω) = Ŵ1 q,Γ (Ω).

KW - Analytic semigroup

KW - Free boundary condition

KW - Maximal L-L regularity

KW - R-Boundedness

KW - Stokes equations

KW - Surface tension

KW - UniformW domain

UR - http://www.scopus.com/inward/record.url?scp=85009758707&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009758707&partnerID=8YFLogxK

U2 - 10.1007/978-4-431-56457-7_9

DO - 10.1007/978-4-431-56457-7_9

M3 - Conference contribution

SN - 9784431564553

VL - 183

SP - 203

EP - 285

BT - Mathematical Fluid Dynamics, Present and Future

PB - Springer New York LLC

ER -