Abstract
AbstractThe Stokes resolvent problem $$\lambda u - \Delta u + \nabla \phi = f$$
λ
u
-
Δ
u
+
∇
ϕ
=
f
with $${\text {div}}(u) = 0$$
div
(
u
)
=
0
subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of $$\mathrm {L}^2_{\sigma } (\Omega ) \ni f \mapsto \phi \in \mathrm {L}^2 (\Omega )$$
L
σ
2
(
Ω
)
∋
f
↦
ϕ
∈
L
2
(
Ω
)
decays like $$|\lambda |^{- 1 / 2}$$
|
λ
|
-
1
/
2
which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like $$|\lambda |^{- \alpha }$$
|
λ
|
-
α
for $$0 \le \alpha \le 1 / 4$$
0
≤
α
≤
1
/
4
and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain $$\Omega $$
Ω
is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side $$f \in \mathrm {L}^2 (\Omega ; {\mathbb {C}}^d)$$
f
∈
L
2
(
Ω
;
C
d
)
admit $$\mathrm {H}^2$$
H
2
-regularity and further prove localized $$\mathrm {H}^2$$
H
2
-estimates for the Stokes resolvent problem. By a generalized version of Shen’s $$\mathrm {L}^p$$
L
p
-extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in $$\mathrm {L}^p (\Omega ; {\mathbb {C}}^d)$$
L
p
(
Ω
;
C
d
)
for $$2d / (d + 2)< p < 2d / (d - 2)$$
2
d
/
(
d
+
2
)
<
p
<
2
d
/
(
d
-
2
)
(with $$1< p < \infty $$
1
<
p
<
∞
if $$d = 2$$
d
=
2
). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.
Funder
Johannes Gutenberg-Universität Mainz
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Reference58 articles.
1. Abels, H.: Boundedness of imaginary powers of the Stokes operator in an infinite layer. J. Evol. Equ. 2(4), 439–457 (2002)
2. Abels, H.: The initial-value problem for the Navier-Stokes equations with a free surface in $$L^q$$-Sobolev spaces. Adv. Differ. Equ. 10(1), 45–64 (2005)
3. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory Grundlehren der mathematischen Wissenschaften, vol. 314. Springer, Berlin (1996)
4. Beale, J.T.: The initial value problem for the Navier-Stokes equations with a free surface. Commun. Pure Appl. Math. 34(3), 359–392 (1981)
5. Blunck, S., Kunstmann, P.C.: Calderón-Zygmund theory for non-integral operators and the $$H^{\infty }$$-functional calculus. Rev. Mater. Iberoamericana 19(3), 919–942 (2003)
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献