Ali Abdennadher, Marie Christine Neel
Estimates for the resolvent of the Stokes operator with periodic boundary conditions in a layer of ${\Bbb R^3}$
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 53, 61-74 (1999)
35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07,
47A10 Spectrum, resolvent
35J45 General theory of elliptic systems of PDE
35J55 Boundary value problems for elliptic systems
Abstract: In this paper, we study the Stokes
operator between two parallel planes of ${\mathbb R}^3$.
We impose Dirichlet homogeneous boundary-conditions, and we
consider vector fields, with fixed period . We show that the semi
group, generated by the Stokes operator, is analytic with respect to the
$L^p$ norm under these boundary conditions.
This property is closely related to an estimate of the
operator's resolvent. For $p=2$, we prove that this resolvent
exists, then that it satisfies the desired estimate. To do this, we use
a decomposition principle of [1], splitting the solenoidal vector fields
of $\mathbb R^3$ into poloidal and toroidal fields and the mean flow.
For larger values of $p$, we show that the estimate, obtained for
$p=2$, extends to larger values of $p$, in view of a result of [2],
valid in a half-space for vectors with prescribed divergence.

Keywords: Stokes and Navier-Stokes equations, Spectrum, resolvent, General theory of elliptic systems of PDE, Boundary value problems for elliptic systems