Boundedness of imaginary powers of the stokes operator in an infinite layer

Abstract.

In this article we prove the existence of bounded purely imaginary powers of the Stokes operator $A_q$, which is defined on the space of solenoidal vector fields $J_q(\Omega), 1$ < q < $\infty$, where $\Omega={\hbox{\opens R}}^{n-1}\times (-1,1)$ is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of the Stokes operator on $\mathbb{R}^n$, a composition of a trace and a Poisson operator – a singular Green operator – and a negligible part.