The Resolvent Problem for the Stokes Equations on Halfspace in $L_p $
Marjorie F. McCracken · SIAM Journal on Mathematical Analysis · 1981
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 \[ | \lambda | \| {\bf u} \|_{L_p(H)} + u \| {\Delta {\bf u}} \|_{L_p(H)} \leqq \| {\bf f} \|_{L_p(H)} \] 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 \[ \begin{gathered} \left. \begin{gathered} \hfill \frac{{\partial {\bf u}}} {{\partial t}}(x,t) - u \Delta _x {\bf u}(x,t) + abla _x p(x,t) = 0, \\ \hfill abla \cdot {\bf u}(x,t) = 0, \\ \end{gathered} \right\}\qquad x \in H, \hfill \\ \left. \qquad \qquad \qquad \qquad {\bf u} \right|_{\partial H} = 0,\,\,\, \hfill \\ \qquad \qquad \qquad \qquad {\bf u}(x,0) = {\bf u}_0 (x) \hfill \\ \end{gathered} \] has a unique solution ${\bf u}$ satisfying the conditions that $\| {\bf u} \|_{L_p (H)} \leqq M\| {u_0 } \|_{L_p (H)} $, that ${\bf u}$ is an analytic function of ${\bf t}$, and other properties of analytic semigroups.