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

Source abstract

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 given ${\bf f} \in L_p (H)$, we find ${\bf u}$, $\nabla _P$ such that \[\begin{gathered} \left. \begin{gathered} \hfill \lambda {\bf u}(x) - \nu \Delta {\bf u}(x) + \nabla _p (x) = {\bf f}(x), \\ \hfill \nabla \cdot u(c) = 0, \\ \end{gathered} \right\}\quad x \in H, \hfill \\ \qquad \qquad \qquad \qquad \left. {\bf u} \right|_{\partial H} = 0 \hfill \\ \end{gathered} \] We show that if $\lambda \nleqq 0$ and if $\nu > 0$ 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 \[ \begin{gathered} \left. \begin{gathered} \hfill \frac{{\partial {\bf u}}} {{\partial t}}(x,t) - \nu \Delta _x {\bf u}(x,t) + \nabla _x p(x,t) = 0, \\ \hfill \nabla \cdot {\bf u}(x,t) = 0, \\ \end{gathered} \right\}\qquad x \in H, \hfill \\ \left. \qquad \qquad \qquad \qquad {\bf u} \right|_{\partial H} = 0,\,\,\, \hfill \\ \qquad \qquad \qquad \qquad {\bf u}(x,0) = {\bf u}_0 (x) \hfill \\ \end{gathered} \] 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.