$ L^p $-Theory of the Stokes equation in a half space

Abstract.

In this paper, we investigate $L^p$-estimates for the solution of the Stokes equation in a half space H where $1\leq p \leq \infty$. It is shown that the solution of the Stokes equation is governed by an analytic semigroup on $BUC_\sigma(H), C_{0,\sigma}(H)$ or $L^\infty_\sigma(H)$. From the operatortheoretical point of view it is a surprising fact that the corresponding result for $L^1_\sigma(H)$ does not hold true. In fact, there exists an $L^1$-function f satisfying ${\it div} f = 0$ such that the solution of the corresponding resolvent equation with right hand side f does not belong to $L^1$. Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the $L^1$-result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded $H^\infty$-calculus on $L^p$ for 1 < p <$\infty$ and obtain as a consequence maximal $L^p-L^q$-regularity for the solution of the Stokes equation.