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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Maximal Regularity of the Stokes Operator in General Unbounded Domains of ℝ n
verfasst von : Reinhard Farwig, Hideo Kozono, Hermann Sohr
Erschienen in: Functional Analysis and Evolution Equations
Verlag: Birkhäuser Basel
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It is well known that the Helmholtz decomposition of
L
q
-spaces fails to exist for certain unbounded smooth domains unless
q
≠ 2. Hence also the Stokes operator and the Stokes semigroup are not well defined for these domains when
q
≠ 2. In this note, we generalize a new approach to the Stokes operator in general unbounded domains from the three-dimensional case, see [
6
], to the
n
-dimensional one,
n
≥ 2, by replacing the space
L
q
, 1 <
q
< ∞, by
$$ \tilde L^q {\text{ where }}\tilde L^q $$
=
L
q
∩
L
2
for
q
≥ 2 and
$$ \tilde L_q $$
=
L
q
+
L
2
for 1 <
q
< 2. As a main result we show that the nonstationary Stokes equation has maximal regularity in
L
8
(0,
T
;
$$ \tilde L_q $$
), 1 <
s, q
< ∞,
T
> 0, for every unbounded domain of uniform
C
1,1
-type in ℝ
n
.