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

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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 -