The Resolvent Problem for the Stokes Equations on Halfspace in $L_p $

The resolvent problem for the Stokes equations on halfspace in $R^3 $ is considered. Letting $H = \{ {(x_1 ,x_2 ,x_3 ) \in R^3 | {x_3 0$ and $1 < p < \infty $, the solution is unique and ${\bf u} \in W^{2,p} $ satisfies \[ | \lambda | \| {\bf u} \|_{L_p(H)} + \nu \| {\Delta {\bf u}} \|_{L_p(H)} \leqq \| {\bf f} \|_{L_p(H)} \] where c depends on p and arg $\lambda $ only.This enables us to prove that the nonstationary Stokes equations generate a bounded analytic semigroup on $L_p (H)$, $1 < p < \infty $. That is, given ${\bf u}_0 \in L_p (H)$, the problem \[ \begin{gathered} \...