### Abstract

In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform W_{r}
^{2-11/r} domain of N-dimensional Euclidean space ℝ^{N} (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σ_{ε,λ0} = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ_{0}}, from which we can deduce the Lp-L_{q} maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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_{q}
^{1}(Ω) to the variational problem:(p, φ)_{Ω} = (f, φ)n for any φ ∈ W _{q}
^{1} (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where W_{q}
^{1}(Ω) is a closed subspace of Ŵ_{qΓ}
^{1}(Ω) = {p ∈ L _{q,loc}(Ω): _{p} ∈ L_{q}(Ω) ^{N}, p|λ = 0} with respect to gradient norm || · || _{Lq}(Ω) that contains a space W_{q}
^{1}(Ω) = {p ∈ W_{q}
^{1}(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with W_{q}
^{1}(Ω) = Ŵ_{q}
^{1}(Ω), and by exterior domains with W _{q}
^{1}(Ω) = the closure of W_{qΓ} ^{1}(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform W_{r}
^{2-1r} domain.

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

Pages (from-to) | 313-368 |

Number of pages | 56 |

Journal | Differential and Integral Equations |

Volume | 27 |

Issue number | 3-4 |

Publication status | Published - 2014 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**On the r-boundedness of solution operators for the stokes equations with free boundary condition.** / Shibata, Yoshihiro.

Research output: Contribution to journal › Article

*Differential and Integral Equations*, vol. 27, no. 3-4, pp. 313-368.

}

TY - JOUR

T1 - On the r-boundedness of solution operators for the stokes equations with free boundary condition

AU - Shibata, Yoshihiro

PY - 2014

Y1 - 2014

N2 - In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of ŴqΓ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of WqΓ 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.

AB - In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform Wr 2-11/r domain of N-dimensional Euclidean space ℝN (N ≥ 2, N < r < ∞). We prove the R-boundedness of solution operators with spectral parameter λ varying in a sector Σε,λ0 = {λ ε C: |arg λ| ≤ π - ε, |λ| ≥ λ0}, from which we can deduce the Lp-Lq maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. 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 ∈ Wq 1(Ω) to the variational problem:(p, φ)Ω = (f, φ)n for any φ ∈ W q 1 (Ω) with 1 < q < ∞ and q′ = q/(q - 1), where Wq 1(Ω) is a closed subspace of ŴqΓ 1(Ω) = {p ∈ L q,loc(Ω): p ∈ Lq(Ω) N, p|λ = 0} with respect to gradient norm || · || Lq(Ω) that contains a space Wq 1(Ω) = {p ∈ Wq 1(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 [28]. Our assumption is satisfied for any q ∈ (l, ∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with Wq 1(Ω) = Ŵq 1(Ω), and by exterior domains with W q 1(Ω) = the closure of WqΓ 1(Ω) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform Wr 2-1r domain.

UR - http://www.scopus.com/inward/record.url?scp=84897503039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897503039&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 313

EP - 368

JO - Differential and Integral Equations

JF - Differential and Integral Equations

SN - 0893-4983

IS - 3-4

ER -