Abstract
In this paper, we proved the generalized resolvent estimate and the maximal Lp-Lq 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 Σε,γ0 = {λ ∈ C \ {0} | |argλ| ≤ π – ε, |λ| ≥ γ0} with 0 < ε < 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 Lp-Lq regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal Lp regularity implies that the resolvent estimate of A in λ ∈ Σε,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 Σε,λ0, we derive a systematic way to prove the resolvent estimate and the maximal Lp regularity at the same time.
