Lp-Theory of the Stokes equation in a half space

In this paper, we investigate L^p-estimates for the solution of the Stokes equation in a half space H where 1 <p <∞. It is shown that the solution of the Stokes equation is governed by an analytic semigroup on B U C_σ(H), C_0,σ(H) or L_σ ^∞(H). From the operatortheoretical point of view it is a surprising fact that the corresponding result for L_σ ¹(H) does not hold true. In fact, there exists an L¹ -function f satisfying div f = 0 such that the solution of the corresponding resolvent equation with right hand side f does not belong to L¹ . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the L¹ -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded H^∞-calculus on L^p for 1 <p <∞ and obtain as a consequence maximal L^p-L^q-regularity for the solution of the Stokes equation.