On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems
On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems
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DOI：
10.2969/jmsj/06420561
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发表时间：
2012-04
影响因子：
0.7
通讯作者：
Y. Shibata;Senjo Shimizu
中科院分区：
文献类型：
--
作者：
Y. Shibata;Senjo Shimizu
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 Σ2,γ0 = {λ ∈ C \ {0} | | arg λ| ≤ π − 2, |λ| ≥ γ0} with 0 < 2 < 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 ∈ Σ2,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 Σ2,λ0 , we derive a systematic way to prove the resolvent estimate and the maximal Lp regularity at the same time.
