  The Stokes resolvent problem $\lambda u - \Delta u + \nabla \phi = f$ with
$\mathrm{div}(u) = 0$ 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
$\mathrm{L}^2_{\sigma} (\Omega) \ni f \mapsto \pi \in \mathrm{L}^2 (\Omega)$
decays like $\lvert \lambda \rvert^{- 1 / 2}$ which agrees exactly with the
scaling of the equation. In comparison to that, we show that the operator norm
of this mapping under Dirichlet boundary conditions decays like $\lvert \lambda
\rvert^{- \alpha}$ for $0 \leq \alpha < 1 / 4$ and we show that this decay rate
cannot be improved to any exponent $\alpha > 1 / 4$, 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. 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 on general Lipschitz domains
and is to the best knowledge of the author the first result that provides
$\mathrm{L}^p$-estimates for the Stokes resolvent subject to Neumann-type
boundary conditions on general convex domains.
