On the maximal L_p-L_q regularity of the Stokes problem with first order boundary condition; model problems

摘要

In this paper, we proved the generalized resolvent estimate and the maximal L_p-L_q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector ∑_(∈,γ0)= {λ ∈ C {0} | |argλ| < π -∈, |λ|≥ γ0} with 0 < ∈ < π/2 and γ0 ≥ 0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R, boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L_p-Lq regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L_p regularity implies that the resolvent estimate of A in λ ∈ ∑_(∈,γ0), but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the R boundedness of the operator family in the sector ∑(∈,λ_0), we derive a systematic way to prove the resolvent estimate and the maximal L_p regularity at the same time.