TY - JOUR

T1 - Generalized resolvent estimates of the stokes equations with first order boundary condition in a general domain

AU - Shibata, Yoshihiro

PY - 2013/3

Y1 - 2013/3

N2 - In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain Ω of the N-dimensional Eulidean space ℝN, N ≥ 2. This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter Λ varying in a sector ∑σ, λ0 = {λ ∈ ℂ |arg λ| < π-σ, |λ| ≥ λ0}, where 0 < σ < π/2 and λ0 ≥ 1. The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution p ∈ Ŵ1q,Γ, (Ω) to the variational problem: (∇p, ∇ φ) = (f, ∇φ) for any φ ∈ Ŵ1q',Γ(Ω). Here, 1 < q < ∞, q' = q/(q-1), Ŵ1q,Γ(Ω) is the closure of Ŵ1q,Γ(Ω) = {p ∈ Ŵ1q(Ω) |p|Γ = 0} by the semi-norm ||∇ ̇ ||Lq(Ω), and Γ is the boundary of Ω. In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in (λ0, ∞). Our assumption is satisfied for any q ∈ (1, ∞) by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.

AB - In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain Ω of the N-dimensional Eulidean space ℝN, N ≥ 2. This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter Λ varying in a sector ∑σ, λ0 = {λ ∈ ℂ |arg λ| < π-σ, |λ| ≥ λ0}, where 0 < σ < π/2 and λ0 ≥ 1. The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution p ∈ Ŵ1q,Γ, (Ω) to the variational problem: (∇p, ∇ φ) = (f, ∇φ) for any φ ∈ Ŵ1q',Γ(Ω). Here, 1 < q < ∞, q' = q/(q-1), Ŵ1q,Γ(Ω) is the closure of Ŵ1q,Γ(Ω) = {p ∈ Ŵ1q(Ω) |p|Γ = 0} by the semi-norm ||∇ ̇ ||Lq(Ω), and Γ is the boundary of Ω. In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in (λ0, ∞). Our assumption is satisfied for any q ∈ (1, ∞) by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.

KW - 76D07

KW - Mathematics Subject Classification (2000): 35Q35

UR - http://www.scopus.com/inward/record.url?scp=84874044309&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874044309&partnerID=8YFLogxK

U2 - 10.1007/s00021-012-0130-1

DO - 10.1007/s00021-012-0130-1

M3 - Article

AN - SCOPUS:84874044309

VL - 15

SP - 1

EP - 40

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

SN - 1422-6928

IS - 1

ER -