Post on 10-Aug-2020
On the Stokes semigroup in some non-Helmholtz domains
Yoshikazu Giga University of Tokyo
June 2014
Joint work with Ken Abe (Nagoya U.), Katharina Schade (TU Darmstadt) and Takuya Suzuki (U. Tokyo)
Contents 1. Introduction
2. Bogovskiโs example and main results
3. Neumann problems with singular data
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1. Introduction Stokes system: ๐ฃ๐ก โ ฮ๐ฃ + ๐ป๐ = 0, div ๐ฃ = 0 in ฮฉ ร (0,๐)
B.C. ๐ฃ = 0 on ๐ฮฉ I.C. ๐ฃ|๐ก=0 = ๐ฃ0 in ฮฉ
Here ฮฉ is a uniformly ๐ถ3-domain in ๐๐(๐ โฅ 2) ๐ฃ : unknown velocity field ๐ : unknown pressure field ๐ฃ0 : a given initial velocity
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Problem 1. Is the solution operator (called the Stokes semigroup) ๐ ๐ก :๐ฃ0 โฆ ๐ฃ(โ , ๐ก) an analytic semigroup in ๐ฟโ-type spaces? In other words, is there ๐ถ > 0 s.t.
๐๐๐ก๐ ๐ก ๐
๐โค๐ถ๐ก๐ ๐, ๐ก โ 0,1 , ๐ โ ๐
where ๐ is an ๐ฟโ-type Banach space? Analyticity is a notion of regularizing effect appeared in parabolic problems in an abstract level.
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Definition of analyticity Definition 1 (semigroup). Let ๐ = ๐ ๐ก ๐ก>0 be a family of bounded linear operators in a Banach space ๐. In other words, ๐ ๐ก ๐ก>0 โ ๐ฟ(๐). We say that ๐ is a semigroup in ๐ if (i) (semigroup property) ๐ ๐ก ๐(๐) = ๐(๐ก + ๐)
for ๐ก, ๐ > 0 (ii) (strong continuity) ๐ ๐ก ๐ โ ๐ ๐ก0 ๐ in ๐ as ๐ก โ ๐ก0 for all ๐ก0 > 0, ๐ โ ๐ (iii) (non degeneracy) ๐ ๐ก ๐ = 0 for all ๐ก > 0
implies ๐ = 0. (iv) (boundedness) ๐ ๐ก ๐๐ โค ๐ถโ for ๐ก โ (0,1)
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Definition 2 (non ๐ถ0 analytic semigroup). Let ๐ be a semigroup in ๐. We say that ๐ is analytic if ๐ถ > 0โ such that
๐๐๐ก๐ ๐ก
๐๐โค๐ถ๐ก
, ๐ก โ 0,1 .
See a book [ABHN] W. Arendt, Ch. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Birkhรคuser (2011).
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Definition 3 (๐ถ0-semigroup). A semigroup ๐ is called ๐ช๐-semigroup if ๐ ๐ก ๐ โ ๐ as ๐ก โ 0 for all ๐ โ ๐.
Remark. The name of analyticity stems from the fact that ๐ = ๐ ๐ก ๐กโฅ0 can be extended as a holomorphic function to a sectorial region of ๐ก i.e. arg ๐ก < ๐ with some ๐ โ (0,๐/2).
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Problem 2. Is the solution operator ๐ ๐ก an analytic semigroup in ๐ฟ๐-type spaces? This problem has a long history. V. A. Solonnilov โ77, Y. G. โ81, โฆโฆ
A classical problem
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Heat semigroup (Gauss โ Weierstrass semigroup)
๐ป ๐ก ๐ ๐ฅ = ๐๐กฮ๐ = ๐บ๐ก โ ๐
= ๏ฟฝ ๐บ๐ก ๐ฅ โ ๐ฆ ๐ ๐ฆ ๐๐ฆ๐๐
๐บ๐ก ๐ฅ =1
4๐๐ก ๐/2 exp(โ ๐ฅ 2/4๐ก)
A simple example
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Proposition 1. (i) The family ๐ป = ๐ป ๐ก ๐ก>0 is a non ๐ถ0-analytic semigroup in ๐ฟโ(๐๐)
(and also in ๐ต๐ถ(๐๐)) but a ๐ถ0-analytic semigroup in ๐ต๐ต๐ถ(๐๐)
(and also in ๐ถ0(๐๐)) (ii) The family ๐ป is a ๐ถ0-analytic semigroup in
๐ฟ๐(๐๐) for all 1 โค ๐ < โ.
๐ต๐ถ ๐๐ = ๐ถ ๐๐ โฉ ๐ฟโ ๐๐ ๐ต๐ต๐ถ ๐๐
= ๐ โ ๐ต๐ถ ๐๐ ๐: uniformly continuous ๐ถ0 ๐๐ = ๐ฟโ-closure of ๐ถ๐โ ๐๐
= ๐ โ ๐ถ ๐๐ lim๐ฅ โโ
๐ ๐ฅ = 0 10
๐ถ๐,๐โ (ฮฉ) = ๐ โ ๐ถ๐โ ฮฉ div ๐ = 0
= the space of all smooth solenoidal vector fields with compact support
๐ถ0,๐(ฮฉ) = ๐ฟโ-closure of ๐ถ๐,๐โ (ฮฉ)
= ๐ โ ๐ถ(ฮฉ๏ฟฝ) div ๐ = 0 in ฮฉ, ๐ = 0 on ๐ฮฉ if ฮฉ is bounded. (Maremonti โ09)
๐ฟ๐๐ (ฮฉ) = ๐ฟ๐-closure of ๐ถ๐,๐โ (ฮฉ), 1 โค ๐ < โ
Spaces for divergence free vector fields
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Helmholtz decomposition ( ฮฉ bounded ๐ถ1-domain, โฆ )
๐ณ๐ ๐ = ๐ณ๐๐ ๐ โจ๐ฎ๐ ๐ 1 < ๐ < โ , where ๐บ๐ ฮฉ = ๐ป๐ โ ๐ฟ๐ ฮฉ ๐ โ ๐ฟ๐๐๐1 (ฮฉ) . ๐ฟ๐๐ ฮฉ = ๐บ๐๐ ฮฉ โฅ
= ๐ โ ๐ฟ๐ ฮฉ ๏ฟฝ ๐ โ ๐ป๐๐๐ฅ = 0ฮฉ
for all ๐ โ ๐บ๐โฒ(ฮฉ)
e.g. Fujiwara โ Morimoto โ79, Galdiโs book โ11
Here 1 ๐โ + 1 ๐โฒโ = 1 ๐ฟ๐โ ฮฉ : = ๐ โ ๐ฟโ(ฮฉ) โซ ๐ โ ๐ป๐๐๐ฅ = 0ฮฉ for all ๐ โ ๐๏ฟฝ 1,1(ฮฉ) .
๐ถ0,๐ ฮฉ โ ๐ต๐ต๐ถ๐ ฮฉ โ ๐ฟ๐โ ฮฉ
More spaces
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Analyticity results for the Stokes semigroup ๐บ ๐ = ๐โ๐๐ in ๐ณ๐๐ (= ๐ณ๐-closure of ๐ช๐,๐
โ ) (i) ๐ฟ๐2 : easy since the Stokes operator is nonnegative
self-adljoint. (ii) ๐ฟ๐๐ : V. A. Solonnikov '77, Y. G. โ81 (bdd domain) (max regularity / resolvent estimate) โฆ H. Abels โ Y. Terasawa โ09 (variable coefficient) bdd, exterior, bent half space.
(iii) ๐ฟ๏ฟฝ๐๐ space = ๏ฟฝ๐ฟ๐ โฉ ๐ฟ๐2 ๐ โฅ 2๐ฟ๐ + ๐ฟ๐2 ๐ < 2
W. Farwig, H. Kozono and H. Sohr โ05, โ07, โ09 General uniformity ๐ถ2-domain / All except
Solonnikov appeals to the resolvent estimate 13
Theorem 1 (M. Geiรert, H. Heck, M. Hieber and O. Sawada, J. Reine Angew. Math., 2012). If a uniformly ๐ถ2 domain ฮฉ admits the Helmholtz decomposition in ๐ฟ๐, then ๐(๐ก) is a ๐ถ0-analytic semigroup in ๐ฟ๐๐ . (Moreover, the maximum ๐ฟ๐regularity holds.)
General results in ๐ณ๐ setting
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Applications to the Navier-Stokes equations โ strong solvability
๐ฃ๐ก โ ฮ๐ฃ + ๐ฃ โ ๐ป๐ฃ + ๐ป๐ = 0, div ๐ฃ = 0 in ฮฉ ร (0,๐) B.C. ๐ฃ = 0 on ๐ฮฉ I.C. ๐ฃ|๐ก=0= ๐ฃ0 in ฮฉ
T. Kato โ H. Fujita โ62: ๐ฟ2 theory, ฮฉ: bdd, ๐ป1/2 initial data โ local existence (3-D) Y. G. โ T. Miyakawa โ85: ๐ฟ๐ theory, ฮฉ: bdd, ๐ฟ๐ initial data โ local existence (๐-D) โฆโฆ H. Amann โ00 Besov theory, โฆโฆ R. Farwig, H. Sohr, W. Varnhorn โ09 โฆโฆ
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Results in ๐ณโ setting Theorem 2 (K. Abe โ Y. G., Acta Math., 2013). Let ฮฉ be a bounded ๐ถ3-domain in ๐๐(๐ โฅ 2). Then the Stokes semigroup ๐(๐ก) is a ๐ถ0 -analytic semigroup in ๐ถ0,๐ ฮฉ (= ๐ต๐ต๐ถ๐ ฮฉ ). It can be regarded as a non ๐ถ0-analytic semigroup in ๐ฟ๐โ(ฮฉ). Remark. Whole space case is reduced to the heat semigroup. This type of analyticity result had been only known for half space where the solution is written explicitly (Desch โ Hieber โ Prรผss โ01, Solonnikov โ03)
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Theorem 3 (K. Abe โ Y. G., J. Evol. Eq., 2012). Let ฮฉ be an ๐ถ3-exterior domain in ๐๐. Then the Stokes semigroup ๐ ๐ก ๐ก>0 is a ๐ถ0 -analytic semigroup in ๐ถ0,๐ ฮฉ and extends to a non ๐ถ0- analytic semigroup in ๐ฟ๐โ(ฮฉ). It can be extended as a ๐ถ0-analytic semigroup in ๐ต๐ต๐ถ๐ ฮฉ . Note that for an unbounded domain ๐ถ0,๐ ฮฉ is strictly smaller than ๐ต๐ต๐ถ๐ ฮฉ because ๐ โ ๐ถ0,๐ ฮฉ implies ๐(๐ฅ) โ 0 as ๐ฅ โ โ.
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Applications to the Navier-Stokes equations โ ๐ณโ theory
Whole space, J. Leray โ34 โฆโฆ Y. G. โ K. Inui โ S. Matsui โ99 Half space, V. A. Solonnikov โ03, H.-O. Bae โ B. J. Jin โ12
๐ ๐ก ๐๐ป๐ โ โค ๐ถ๐กโ1 2โ ๐ โ
Bounded and exterior domains by Ken Abe โ14:
๐ ๐ก ๐๐ป๐ โ โค ๐ถ๐กโ๐ผ2 ๐ โ
๐ผ ๐ป๐ โ1โ๐ผ , 0 < ๐ผ < 1
For strong solutions of N-S,
๐ : existence time ๐ โฅ ๐ถ๐ฃ0 โ
2
๐โ: possible blow-up time ๐ฃ(๐ก) โ โฅ ๐ถ/ ๐โ โ ๐ก 1 2โ 18
(i) 2nd order operator on ๐ (one dim): K. Yosida โ66 (ii) 2nd order elliptic operator K. Masuda โ71, โ72 book in โ75 ๐ฟ๐ theory, cutoff procedure for resolvent and interpolation (iii) higher order, H. B. Stewart โ74, โ80 Masuda โ Stewart method (iv) degenerate + mixed B. C. K. Taira, โ04 See also: P. Acquistapace, B. Terrani (1987) A. Lunardi (1995) Book. More recent. nonsmooth coefficient / nonsmooth domain Heck โ Hieber โ Stavarakidis (2010) VMO coeff., higher order Arendt โ Schaetzle (2010) 2nd order, Lipschitz domain Takuya Suzuki (2014) ๐ถ1-domain, any order.
๐ณโ-theory for ๐โ๐๐ณ where ๐ณ is an elliptic operator
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Extensions of ๐ณโ-theory (i) Theorems 2 and 3 are obtained by a direct
analysis of semigroup. It applies to a perturbed half space (K. Abe).
(ii) Resolvent estimate is obtained by extending a method of Masuda โ Stewart (Maximum analyticity angle is obtained, K. Abe, Y. G. and M. Hieber, to appear.)
(iii) Cylindrical domains are OK. work in progress (K. Abe, Y. G., K. Schade, T. Suzuki)
(iv) Counterexample for a layer domain 0 < ๐ฅ๐ < 1 for ๐ โฅ 3. (L. von Below) 20
We know if ๐ฟ๐ admits the Helmholtz decomposition, ๐ ๐ก is analytic in ๐ฟ๐๐ . Is the Helmholtz decomposition really necessary to conclude that ๐ ๐ก is analytic in ๐ฟ๐๐ ? (If ๐ = 2, the Helmholtz decomposition always exists and the Stokes operator is self-adjoint so that ๐ ๐ก is always analytic no matter what ฮฉ is.)
Problem
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The Helmholtz decomposition in ๐ฟ๐ is not a necessary condition so that ๐ ๐ก is a ๐ถ0 analytic semigroup in ๐ฟ๐๐ ฮฉ .
Our answer
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2. Bogovskiโs example and main results
0 ๐ ๐๐
๐ฅ1
๐๐ = ๐ฅ = (๐ฅ1, ๐ฅ2) arg ๐ฅ < ๐ 2โ We say that a planar domain ฮฉ is a sector-like domain with opening angle 0 < ๐ < 2๐ if
ฮฉ โ ๐ท๐ = ๐๐ โ ๐ท๐
for some ๐ > 0, where ๐ท๐ is an open disk of radius ๐ centered at the origin.
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Bogovskiโs example
๐ ฮฉ
๐ฅ1
Proposition 2. The ๐ฟ๐ Helmholtz decomposition (๐ > 2) fails for a smooth sector-like domain when ๐ > ๐ โ (1 โ 2/๐).
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Note that for 4 3โ < r < 4, ๐ฟ๐ Helmholtz decomposition holds for all ๐ โค ๐ < 2๐.
Main results Theorem 4 [AGSS]. Let ฮฉ (โ ๐2) be a ๐ถ3 sector-like domain. Then the Stokes semigroup ๐(๐ก) is a ๐ถ0-analytic semigroup in ๐ฟ๐๐ ฮฉ for any 2 โค ๐ < โ. This follows from interpolation with ๐ฟ2 result and the following ๐ฟโ-result. Theorem 5 [AGSS]. Let ฮฉ be a ๐ถ3 sector-like domain. Then ๐(๐ก) is a ๐ถ0-analytic semigroup in ๐ถ0,๐ ฮฉ . (Moreover, ๐ก ๐ป2๐ ๐ก ๐ฃ0 โ โค ๐ถ๐ ๐ฃ0 โ for ๐ก โ (0,๐).) AGSS = K. Abe, Y. G., K. Schade, T. Suzuki 25
Idea of the proof โ a blow-up argument a key observation
(Harmonic) pressure gradient estimate by velocity gradient: sup๐ฅโฮฉ
๐ฮฉ(๐ฅ) ๐ป๐(๐ฅ, ๐ก) โค ๐ถ ๐ป๐ฃ ๐ฟโ ๐ฮฉ ๐ก
๐ฮฉ ๐ฅ = dist (๐ฅ,๐ฮฉ) .
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Bounds of ๐ป๐ is not enough to guarantee the uniqueness. Parasitic solution: ๐ฃ = ๐ ๐ก , ๐ ๐ฅ, ๐ก = โ๐๐ ๐ก โ ๐ฅ in ๐๐. Poiseuille flow type: half space
๐ฃ = ๐ฃ1 ๐ฅ๐ , 0, โฆ , 0 ,ใ๐ = ๐ ๐ก ๐ฅ1
๏ฟฝ ๐๐ก โ ฮ ๐ฃ1 = โ๐ ๐ก ๐ฃ1 = 0 on ๐ฅ๐ = 0
(div ๐ฃ = 0 is automatic)
Pressure should be related to velocity
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Consider
๐ฃ๐ก โ ฮ๐ฃ + ๐ป๐ = 0 in ฮฉ. Take divergence to get
ฮ๐ = 0 in ฮฉ since div ๐ฃ = 0. Take inner product with ๐ฮฉ (unit exterior normal) and use ๐ฃ๐ก โ ๐ฮฉ = 0 to get
๐๐ ๐๐ฮฉโ = ๐ฮฉ โ ฮ๐ฃ on ๐ฮฉ.
Equations for the pressure
3. Neumann problems with singular data
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Lemma 1. If div ๐ฃ = 0, then ๐ฮฉ โ ฮ๐ฃ = div๐ฮฉ๐(๐ฃ)
with ๐ ๐ฃ = โ ๐ป๐ฃ โ ๐ป๐ฃ๐ก โ ๐ฮฉ.
In three dimensional case,
๐ฮฉ โ ฮ๐ฃ = โdiv๐ฮฉ ๐ ร ๐ฮฉ where ๐ = curl ๐ฃ (vorticity). In any case ๐ is a tangent vector field.
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The pressure solves
(NP) ฮ๐ = 0 in ฮฉ
๐๐ ๐๐ฮฉโ = div๐ฮฉ๐ on ๐ฮฉ.
Enough to prove that ๐ฮฉ๐ป๐ โ โค ๐ถ ๐ โ
for all tangential vector field ๐.
Neumann problem
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Definition 4 (Weak solution of (NP)). (Ken Abe โ Y. G., โ12) Let ฮฉ be a domain in ๐๐ (๐ โฅ 2) with ๐ถ1 boundary. We call ๐ โ ๐ฟ๐๐๐1 (ฮฉ๏ฟฝ) a weak solution of (NP) for ๐ โ ๐ฟโ(๐ฮฉ) with ๐ โ ๐ฮฉ = 0 if ๐ with ๐ฮฉ๐ป๐ โ ๐ฟโ(ฮฉ) fulfills
๏ฟฝ ๐ฮ๐๐๐ฅฮฉ
= ๏ฟฝ ๐ โ ๐ป๐๐โ๐โ1
๐ฮฉ
for all ๐ โ ๐ถ๐2(ฮฉ๏ฟฝ) satisfying ๐๐ ๐๐ฮฉโ = 0 on ๐ฮฉ.
Strictly admissible domain
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Definition 5 (Strictly admissible domain). Let ฮฉ be a uniformly ๐ถ1 domain. We say that ฮฉ is strictly admissible if there is a constant ๐ถ such that
๐ฮฉ๐ป๐ โ โค ๐ถ ๐ ๐ฟโ(๐ฮฉ)
holds for all weak solution of (NP) for tangential vector fields ๐. Note that strictly admissibility implies admissibility defined below.
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Let ๐: ๐ฟ๏ฟฝ๐ ฮฉ โ ๐ฟ๏ฟฝ๐๐ ฮฉ be the Helmholtz projection and ๐ = ๐ผ โ ๐. Applying ๐ to the Stokes equation to get
๐ป๐ = ๐ ฮ๐ฃ .
Here ๐ฟ๏ฟฝ๐ = ๐ฟ๐ โฉ ๐ฟ2, ๐ฟ๏ฟฝ๐๐ = ๐ฟ๐๐ โฉ ๐ฟ2 for ๐ > 2.
Admissible domain
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Definition 6 (Ken Abe โ Y. G., Acta Math., 2013). Let ฮฉ be a uniformly ๐ถ1-domain. We say that ฮฉ is admissible if there exists ๐ โฅ ๐ and a constant ๐ถ = ๐ถฮฉ such that
sup๐ฮฉ(๐ฅ) ๐ ๐ป โ ๐ (๐ฅ) โค ๐ถ ๐ ๐ฟโ ๐ฮฉ
hold for all matrix value ๐ = ๐๐๐ โ ๐ถ1(ฮฉ๏ฟฝ) satisfy ๐ป โ ๐ = โ ๐๐๐๐๐๐ โ ๐ฟ๏ฟฝ๐ ฮฉ ,
tr ๐ = 0 and ๐โ๐๐๐ = ๐๐๐๐โ for all ๐, ๐, โ = 1, โฆ ,๐ .
Admissible domain (continued)
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Remark. (i) This is a property of the solution of the Neumann problem for the Laplace operator. In fact, ๐ป๐ = ๐ ๐ป โ ๐ is formally equivalent to
โฮ๐ = div(๐ป โ ๐) in ฮฉ ๐๐ ๐๐ฮฉโ = ๐ฮฉ โ (๐ป โ ๐) on ๐ฮฉ.
Under the above condition for ๐ we see that ๐ is harmonic in ฮฉ since
div(๐ป โ ๐) = ๏ฟฝ๐๐๐๐๐๐๐๐,๐
= ๏ฟฝ๐๐๐๐๐๐๐ = 0.
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(ii) The constant ๐ถฮฉ depends on ฮฉ but independent of dilation, translation and rotation.
(iii) If ฮฉ is admissible, we easily obtain the pressure gradient estimate by taking ๐๐๐ = ๐๐๐ฃ๐.
(iv) It turns out that ๐๐ ๐๐ฮฉโ = div๐ฮฉ(๐ฮฉ โ ๐ โ ๐๐ก ).
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Remark. Strictly admissibility implies admissibility. Example of strictly admissible domains
(a) half space (b) ๐ถ3 bounded domain (c) ๐ถ3 exterior domain
Note that layer domain ๐ < ๐ฅ๐ < ๐ is not strictly admissible. Consider ๐ ๐ฅ1, โฆ , ๐ฅ๐ = ๐ฅ1. Theorem 6 (Ken Abe โ Y. G., Acta Math., 2013). If ฮฉ is ๐ถ3 and admissible, then ๐(๐ก) is a ๐ถ0-analytic semigroup in ๐ถ0,๐(ฮฉ). (The conclusion of Theorem 5 holds.)
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Estimates for harmonic pressure gradient
Theorem 7 (A key step). If ฮฉ is a ๐ถ2 sector-like domain, then it is admissible (not strictly admissible). A strictly admissibility is proved for a bounded domain (K. Abe โ Y. G.), exterior domain (K. Abe โ Y. G.), a perturbed half space (K. Abe). For a bounded domain there is another proof by C. Kenig, F. Lin, Z. Shen (2013).
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Sector-like domain
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Lemma 2. Let ฮฉ be a ๐ถ2 sector-like domain in ๐2. Then there is a constant ๐ถ such that
๐ฮฉ ๐ฅ ๐ป๐ข โ โค ๐ถ ๐ โ for all weak solution ๐ข โ ๐ฟ๐๐๐1 ฮฉ๏ฟฝ๐ of (NP) ฮ๐ข = 0 in ฮฉ๐
๐๐๐๐ฮฉ
= div๐ฮฉ ๐ on ๐ฮฉ๐
satisfying ๐ โ ๐ฮฉ = 0 on ๐ฮฉ โฉ ๐ท2๐ and ๐ = 0 on ๐๐ท2๐ โฉ ฮฉ provided that ๐ฮฉ๐ป๐ข โ < โ.
Here ฮฉ๐ โ ฮฉ โฉ ๐ท2๐ .
2๐ ฮฉ๐
๐ฅ1
Note that ๐ถ is independent of ๐ . This estimate yields
Lemma 3. There exists a constant ๐ถ such that all weak solutions ๐ข โ ๐ฟ๐๐๐1 ฮฉ๏ฟฝ of (NP) with ๐๐ โ๐ณ๐ ๐ and ๐ โ ๐ฟโ ๐ฮฉ with ๐ โ ๐ฮฉ = 0 fulfills ๐ฮฉ๐ป๐ข โ โค ๐ถ ๐ โ.
This yields Theorem 7.
Sector-like domain (continued)
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2๐
ฮฉ๐
๐ฅ1
We argue by contradiction ๐ข๐,๐๐,๐ ๐ ๐=1โโ , such
that 1 = ๐ฮฉ๐ป๐ข๐ ๐ฟโ(ฮฉ๐ ๐) > ๐ ๐๐ ๐ฟโ (๐ฮฉโฉ๐ท2๐ ๐)
Case A ๐ ๐ โ โ as ๐ โ โ
Case B lim๐โโ ๐ ๐ < โ
We only discuss case A. We take ๐ฅ๐ โ ฮฉ๐ ๐ such that
๐ฮฉ ๐ฅ๐ ๐ป๐ข๐(๐ฅ๐) > 1 2โ . We may assume ๐ข๐ ๐ฅ๐ = 0 by subtracting a constant.
Idea of the proof of Lemma 2
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Case 1 subsequence ๐ฅ๐๐โ โ ๐ฅ๏ฟฝ
Case 2 ๐ฅ๐ โ โ Case 1 (a) ๐ฅ๏ฟฝ โ ฮฉ, (Case 1 (b) ๐ฅ๏ฟฝ โ ๐ฮฉ)
๐ข๐ converges to ๐ข locally uniformly with its derivatives in ฮฉ and ๐ข ๐ฅ๏ฟฝ = 0. We now apply the uniqueness to conclude ๐ข โก 0 which contradicts ๐ ๐ฅ๏ฟฝ ๐ป๐ข(๐ฅ๏ฟฝ) โฅ 1 2โ .
Compactness
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Lemma 4. Let ฮฉ be a ๐ถ2 sector-like domain. Let ๐ข โ ๐ถ2 ฮฉ โฉ ๐ถ1(ฮฉ๏ฟฝ) be a solution of (NP) in ฮฉ with ๐ = 0. Assume that ๐ฮฉ๐ป๐ข โ < โ. Then ๐ข is a constant function.
Uniqueness
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Case 1 (b) can be handled by uniqueness in a half space with blow-up argument ๐ฃ๐ ๐ฅ = ๐ข๐ ๐ฅ๐ + ๐๐๐ฅ , ๐๐= ๐ฮฉ ๐ฅ๐ .
Case 2 can be treated by scaling-down argument
๐ค๐ ๐ฅ = ๐ข๐ ๐ฅ ๐ฅ๐โ and uniqueness (with zero flux condition) in ๐๐ โฉ ๐ท๐ with some ๐ > 0 or ๐๐.
Other cases
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ฮ๐
๐
(1) Assumption implies zero flux condition
๏ฟฝ๐๐ข๐๐
ฮ๐ โฉฮฉ
๐โ1 = 0, ๐ โซ 1,
where ฮ๐ = ๐๐ท๐ . This implies that ๐ข is bounded in ฮฉ.
A key observation for uniqueness
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(2) We may assume โซ ๐ขฮ๐ โฉฮฉ๐โ1 = 0 for
๐ โซ 1. If ๐ข attains its maximum, the strong maximum principle implies ๐ข = const.
(3) In the case max is not attained, we consider sequence ๐ฅ๐ such that ๐ข ๐ฅ๐ โ sup๐ข , ๐ฅ๐ โ โ . We scale down and obtain a contradiction to the uniqueness result in a sector under zero flux condition.
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Summary โข We prove that a ๐ถ2 sector-like domain is admissible.
โข A similar idea works for a domain with finitely many cylindrical outlets.
Theorem 4. Let ฮฉ (โ ๐2) be a ๐ถ3 sector-like domain. Then the Stokes semigroup ๐(๐ก) is a ๐ถ0 -analytic semigroup in ๐ฟ๐๐ ฮฉ for any 2 โค ๐ < โ.
Theorem 5. Let ฮฉ be a ๐ถ3 sector-like domain. Then ๐(๐ก) is a ๐ถ0-analytic semigroup in ๐ถ0,๐ ฮฉ . (Moreover, ๐ก ๐ป2๐ ๐ก ๐ฃ0 โ โค ๐ถ๐ ๐ฃ0 โ for ๐ก โ (0,๐).)
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