The Stokes resolvent problem \(\lambda u - \Delta u + \nabla \phi = f\) with \({\text {div}}(u) = 0\) subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of \(\mathrm {L}^2_{\sigma } (\Omega ) \ni f \mapsto \phi \in \mathrm {L}^2 (\Omega )\) decays like \(|\lambda |^{- 1 / 2}\) which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like \(|\lambda |^{- \alpha }\) for \(0 \le \alpha \le 1 / 4\) and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain \(\Omega \) is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side \(f \in \mathrm {L}^2 (\Omega ; {\mathbb {C}}^d)\) admit \(\mathrm {H}^2\)-regularity and further prove localized \(\mathrm {H}^2\)-estimates for the Stokes resolvent problem. By a generalized version of Shen’s \(\mathrm {L}^p\)-extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in \(\mathrm {L}^p (\Omega ; {\mathbb {C}}^d)\) for \(2d / (d + 2)< p < 2d / (d - 2)\) (with \(1< p < \infty \) if \(d = 2\)). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.

斯托克斯预解问题\（\拉姆达Ú - \德尔塔U + \ nabla \披= F \）与\（{\文本{DIV}}（U）= 0 \）受到均匀狄利克雷或均相诺伊曼型边界条件被调查。在本文的第一部分中，我们表明对于Neumann型边界条件，\（\ mathrm {L} ^ 2 _ {\ sigma}（\ Omega）\ ni f \ mapsto \ phi \ in \ mathrm {L } ^ 2（\ Omega）\）的衰减像\（| \ lambda | ^ {-1/2} \）一样，它与等式的比例完全一致。与此相比，在Dirichlet边界条件下，此映射的算子范数对于\（0 \ le \ alpha \ le 1/4 \）像\（| \ lambda | ^ {-\ alpha} \）衰减并且我们展示了此速率的最优性，从而违反了方程的自然定标。在本文的第二部分中，如果基础域\（\ Omega \）是凸的，我们将研究在齐次Neumann型边界条件下的斯托克斯分解问题。调用Grisvard著名的结果（在非光滑域椭圆问题的专着和在数学研究中，皮特曼，1985年），我们表明，弱解ü用右手边\（F \中\ mathrm {L} ^ 2（\欧米茄; {\ mathbb {C}} ^ d）\）接受\（\ mathrm {H} ^ 2 \） -正则性，并进一步证明局部\（\ mathrm {H} ^ 2 \）-估计了Stokes解析问题。通过沉氏\（\ mathrm {L} ^ p \）的广义版本外推定理（Shen in Ann Inst Fourier（Grenoble）55（1）：173–197，2005），我们在\（\ mathrm {L} ^ p（\ Omega; {\ mathbb {C }} ^ d）\）为\（2D /（d + 2）<p <2D /（d - 2）\） （用\（1 <p <\ infty \）如果\（d = 2 \） ）。此间隔大于在一般Lipschitz域上受Dirichlet边界条件（Shen in Arch Ration Mech Anal 205（2）：395–424，Shen）约束的可分辨估计的已知间隔。