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/rr 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 ∈ W1q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1q (Ω) is a closed subspace of Ŵ1q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq(Ω) that contains a space W1q,Γ (Ω) = {p ∈ W1q (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 W1q (Ω) = Ŵ1q,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/rr 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 ∈ W1q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1q′(Ω) with 1 < q < ∞ and q′ = q/(q − 1), where W1q (Ω) is a closed subspace of Ŵ1q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq(Ω) that contains a space W1q,Γ (Ω) = {p ∈ W1q (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 W1q (Ω) = Ŵ1q,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

AN - SCOPUS:85009758707

SN - 9784431564553

T3 - Springer Proceedings in Mathematics and Statistics

SP - 203

EP - 285

BT - Mathematical Fluid Dynamics, Present and Future

A2 - Shibata, Yoshihiro

A2 - Suzuki, Yukihito

PB - Springer New York LLC

T2 - 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014

Y2 - 11 November 2014 through 14 November 2014

ER -