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Published in:
2005 | OriginalPaper | Chapter
Bounded Imaginary Powers and H∞-Calculus of the Stokes Operator in Unbounded Domains
Author : H. Abels
Published in: Nonlinear Elliptic and Parabolic Problems
Publisher: Birkhäuser Basel
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In the present contribution we study the Stokes operator
A
q
= -
P
q
Δ on
L
σ
q
(Ω), 1 <
q
< ∞, where Ω is a suitable bounded or unbounded domain in ℝ
n
,
n
≥ 2, with
C
1,1
-boundary. We present some conditions on Ω and the related function spaces and basic equations which guarantee that
c
+
A
q
for suitable
c
∈ ℝ is of positive type and admits a bounded
H
∞
- calculus. This implies the existence of bounded imaginary powers of
c
+
A
q
. Most domains studied in the theory of Navier-Stokes like, e.g., bounded, exterior, and aperture domains as well as asymptotically flat layers satisfy the conditions. The proof is done by constructing an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. Finally, the result is used to proof the existence of a bounded
H
∞
-calculus of the Stokes operator Aq on an aperture domain.