On the maximal Lp-Lq regularity of the Stokes problem with first order boundary condition : model problems

DOI Web Site 被引用文献16件 参考文献95件
  • Shibata Yoshihiro
    Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University
  • Shimizu Senjo
    Department of Mathematics, Faculty of Science, Shizuoka University

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タイトル別名
  • On the maximal <i>L</i><sub><i>p</i></sub>-<i>L</i><sub><i>q</i></sub> regularity of the Stokes problem with first order boundary condition; model problems

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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,γ<sub>0</sub> = {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,γ<sub>0</sub>, 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,λ<sub>0</sub>, we derive a systematic way to prove the resolvent estimate and the maximal Lp regularity at the same time.

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