On the Stokes semigroup in some non-Helmholtz domains

Abe, Ken; Giga, Yoshikazu; Schade, Katharina; Suzuki, Takuya

doi:10.1007/s00013-015-0729-6

Cited by 16 publications

(17 citation statements)

References 13 publications

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“…It turns out that the proof of [AGSS,Theorem 3.2] should be modified with slight modification of the statement. We shall give its rigorous statement.…”

Section: Admissibilitymentioning

confidence: 99%

On the Stokes resolvent estimates for cylindrical domains

et al. 2016

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Abstract. This paper studies the analyticity of the Stokes semigroup in an infinite cylinder or more generally a cylindrical domain with several exits to infinity in the space C0,σ, the L ∞ -closure of all smooth compactly supported solenoidal vector fields. These domains are not strictly admissible in the sense of the first two authors (2014). However, it is shown that these domains are still admissible which yields the analyticity in C0,σ. A new proof based on a blow-up argument is given to derive an L ∞ -type resolvent estimate which enables us to conclude that the analyticity angle of the Stokes semigroup in C0,σ is π/2. Mathematics Subject Classification (2010). Primary 35Q35; Secondary 76D07.

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“…It turns out that the proof of [AGSS,Theorem 3.2] should be modified with slight modification of the statement. We shall give its rigorous statement.…”

Section: Admissibilitymentioning

confidence: 99%

On the Stokes resolvent estimates for cylindrical domains

et al. 2016

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“…In this paper, as a continuation of , and , we study the Stokes semigroup, i.e., the solution operator

S false(t false) : v_{0} \mapsto v false(\cdot, t false)

of the initial‐boundary problem for the Stokes system

v_{t} - Δ v + \nabla q = 0, div v = 0 in Ω \times false(0, \infty false)

with the zero boundary condition

v = 0 on \partial Ω \times (0, \infty)

and the initial condition

{v |}_{t = 0} = v_{0}

, where Ω is a domain in

R^{n}

with

n \geq 2

. It is by now well‐known that

S (t)

forms a C 0 ‐analytic semigroup in

L_{σ}^{p} false(1 < p < \infty false)

for various domains like smooth bounded domains (, ).…”

Section: Introductionmentioning

confidence: 99%

“…We are tempted to interpolate the

L^{\infty}

type result obtained in with the L 2 ‐result. In fact, in the estimates and with

p = \infty

are established for all

v_{0} \in C_{0, σ} false(normalΩ false)

, the

L^{\infty}

‐closure of

C_{c, σ}^{\infty} false(normalΩ false)

for a C 2 sector‐like domain Ω in

R^{2}

.…”

Section: Introductionmentioning

confidence: 99%

“…We are tempted to interpolate the

L^{\infty}

type result obtained in with the L 2 ‐result. In fact, in the estimates and with

p = \infty

are established for all

v_{0} \in C_{0, σ} false(normalΩ false)

, the

L^{\infty}

‐closure of

C_{c, σ}^{\infty} false(normalΩ false)

for a C 2 sector‐like domain Ω in

R^{2}

. However, it is not clear that the complex interpolation space

{[L_{σ}^{2}, C_{0, σ}]}_{ρ}

agrees with

L_{σ}^{p}

with

2 / p = 1 - ρ

although it is well‐known as the Riesz–Thorin theorem that

{([] L^{2}, L^{\infty})}_{ρ} = L^{p}

.…”

Section: Introductionmentioning

confidence: 99%

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On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Bolkart

Giga

Miura

et al. 2017

Mathematische Nachrichten

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“…Masuda and Stewart proved similar analyticity results in C 0 for general elliptic operators , . Further admissible domains are bounded domains , exterior domains , bent half‐spaces , and sector‐like domains in which the Helmholtz projection may not exist . Abe derived from these results on the Stokes equations in

L_{\infty}

existence results for the Navier–Stokes equations in

L_{\infty}

, .…”

Section: Introductionmentioning

confidence: 88%

The Stokes equations in layer domains on spaces of bounded and integrable functions

Below

Bolkart

2016

Mathematische Nachrichten

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The Stokes equations in a layer domain are investigated. For the case of a two‐dimensional layer it is shown by resolvent estimates that there exists a semigroup of angle π/2 on solenoidal subspaces of L∞ and L1. Furthermore, the stationary problem for dimension two is solvable in these spaces. It is also shown that both results cannot be extended to three and higher dimensions in the same setting.

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On the Stokes semigroup in some non-Helmholtz domains

Cited by 16 publications

References 13 publications

On the Stokes resolvent estimates for cylindrical domains

On the Stokes resolvent estimates for cylindrical domains

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

The Stokes equations in layer domains on spaces of bounded and integrable functions

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