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)
MSC:
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