Fluid flow and rotation: a fascinating interplay [-3mm]j j \ layer thickness ". Jurgen Saal (HHU...
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Fluid flow and rotation: a fascinating interplay
Jurgen Saal
Mathematics for Nonlinear Phenomena: Analysis and Computation
International Conference in honor of
Professor Yoshikazu Giga
on his 60th birthday
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 1 / 21
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About 12 years ago in Sapporo there was ...
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... some years later ...
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Contents
1 Rotating fluids
2 The Taylor-Proudman theorem
3 The Ekman boundary layer problem
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Contents
1 Rotating fluids
2 The Taylor-Proudman theorem
3 The Ekman boundary layer problem
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The Ekman boundary layer problem (EBLP)
Important for: daily weather forecast, tornado and hurricane evolution, aviation,pollen distribution, dew, fog, frost forecasts, air pollution, ....
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Modeling of the problem
Navier-Stokes equations with Coriolis term
(EBLP)
∂tv − ν∆v + (v ,∇)v + Ωe3 × v = −∇p
div v = 0
v |∂G = UE |∂Gv |t=0 = v0
considered in G = (R2 × (0, d))× (0,T ) for d ∈ [δ,∞].
The Ekman spiral solution
UE (x3) = U∞(
1− e−x3/δ cos(x3/δ), e−x3/δ sin(x3/δ), 0)
is an exact solution of (EBLP) with pressure pE (x2) = −ΩU∞x2.
Here δ =
√2ν
|Ω|“ layer thickness ”.
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LiteratureGeophysical:
V.W. Ekman, On the influence of the earth’s rotation on ocean currents,Arkiv Matem. Astr. Fysik, 1905.
J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, 1987.
H.P. Greenspan, The Theory of Rotating Fluids, Cambridge Univ. Pr., 1968.
Mathematical:
D. Lilly ’66, J. Atmospheric Sci.
E. Grenier and N. Masmoudi ’97, Commun. Partial. Differ. Equations.
B. Desjardins, E. Dormy, and E. Grenier ’99, Nonlinearity.
J.-Y. Chemin, B. Desjardin, I. Gallagher, and Grenier E., Ekman boundarylayers in rotating fluids ’02, ESAIM Control Optim. Calc. Var.
and Rousset, Greenberg, Marletta, Tretter, Giga, Mahalov, Hess, Hieber,Koba, S., ...
In infinite energy space B0∞,1(R2, Lp((0, d))):
Giga, Inui, Mahalov, Matsui, S. ’07, ARMA.
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Fluid flow about a rotating obstacle
Mathematical model:
∂tu + (u · ∇)u −∆u + Ωe3 × u − Ω(e3 × ξ)∇u = ∇p
div u = 0
u|t=0 = u0
ξ = (x1, x2, x3).
Literature:
T. Hishida ’99, ARMA (iniciated by P.G. Galdi).
and Galdi , Farwig, Muller, Shibata, Neustupa, Necasova, Hieber, Geißert,Heck, Dintelmann, Penel, Disser, Maekawa, ...
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Rotating obstacle vs Ekman
’Rotating obstacle’ situation (e.g. propeller)
additional forces:
Ωe3 × u︸ ︷︷ ︸Corolis
+Ω2
2e3 × (e3 × ξ)︸ ︷︷ ︸centrifugal
− Ω(e3 × ξ)∇u︸ ︷︷ ︸drift
Ω: twice angular velocity of rotation.
’Ekman’ situation
(e.g. Ekman, tornado-hurricane, spin-coating)
additional forces:
Ωe3 × u︸ ︷︷ ︸Corolis
+Ω2
2e3 × (e3 × ξ)︸ ︷︷ ︸centrifugal
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Linearized spectrum
without rotation: Au = P∆u
rotating obstacle: AOu = P(∆u − Ωe3 × u − Ω(e3 × ξ)∇u)
Ekman: AEu = P(∆u − Ωe3 × u − (UE · ∇)u − u3∂3UE )
Spectra:
=⇒
A generates bounded analytic C0-semigroup
AO generates bounded C0-semigroup, which is not analytic
AE generates analytic C0-semigroup, which is not bounded
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Linearized spectrum
without rotation: Au = P∆u
rotating obstacle: AOu = P(∆u − Ωe3 × u − Ω(e3 × ξ)∇u)
Ekman: AEu = P(∆u − Ωe3 × u − (UE · ∇)u − u3∂3UE )
Spectra:
=⇒
A generates bounded analytic C0-semigroup
AO generates bounded C0-semigroup, which is not analytic
AE generates analytic C0-semigroup, which is not bounded
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21
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Linearized spectrum
without rotation: Au = P∆u
rotating obstacle: AOu = P(∆u − Ωe3 × u − Ω(e3 × ξ)∇u)
Ekman: AEu = P(∆u − Ωe3 × u − (UE · ∇)u − u3∂3UE )
Spectra:
=⇒
A generates bounded analytic C0-semigroup
AO generates bounded C0-semigroup, which is not analytic
AE generates analytic C0-semigroup, which is not bounded
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21
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Contents
1 Rotating fluids
2 The Taylor-Proudman theorem
3 The Ekman boundary layer problem
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The Taylor-Proudman theorem
Theorem (after Taylor ’16 and Proudman ’17)
Within a fluid that is steadily rotated at high angular velocity Ω, there is novariation of the velocity field in the direction parallel to the axis of rotation.
(NSC )
∂tu + (u · ∇)u −∆u + Ωe3 × u +∇p = 0
div u = 0
Heuristically:
Ω >> 1 ⇒ Ωe3 × u ≈ −∇p
⇒ ∇× (e3 × u) ≈ 0
⇒ d
dzu = e3 · ∇u ≈ 0
⇒ u(x , y , z) ≈ u(x , y)
⇒ flow is two-dimensional ⇒ global well-posedness !
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21
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The Taylor-Proudman theorem
Theorem (after Taylor ’16 and Proudman ’17)
Within a fluid that is steadily rotated at high angular velocity Ω, there is novariation of the velocity field in the direction parallel to the axis of rotation.
(NSC )
∂tu + (u · ∇)u −∆u + Ωe3 × u +∇p = 0
div u = 0
Heuristically:
Ω >> 1 ⇒ Ωe3 × u ≈ −∇p
⇒ ∇× (e3 × u) ≈ 0
⇒ d
dzu = e3 · ∇u ≈ 0
⇒ u(x , y , z) ≈ u(x , y)
⇒ flow is two-dimensional ⇒ global well-posedness !
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21
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The Taylor-Proudman theorem
Theorem (after Taylor ’16 and Proudman ’17)
Within a fluid that is steadily rotated at high angular velocity Ω, there is novariation of the velocity field in the direction parallel to the axis of rotation.
(NSC )
∂tu + (u · ∇)u −∆u + Ωe3 × u +∇p = 0
div u = 0
Heuristically:
Ω >> 1 ⇒ Ωe3 × u ≈ −∇p
⇒ ∇× (e3 × u) ≈ 0
⇒ d
dzu = e3 · ∇u ≈ 0
⇒ u(x , y , z) ≈ u(x , y)
⇒ flow is two-dimensional ⇒ global well-posedness !
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21
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Literature
After more than 80 years first analytical verification:
Periodic domains (e.g. torus):I Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ.
Math. J., ’03 Russian Math. Surveys
Crucial pre-condition: uniformness in Ω of local results!
Whole space Rn:I J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math.
Appl.I and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ...
Crucial ingredients:I exp(−t(∆ + PΩe3×)) = exp(−t∆) exp(tPΩe3×),
I dispersive estimates:
∣∣∣∣∫R3
exp(ix · ξ + iΩξ3/|ξ|)ψ(ξ) dξ
∣∣∣∣ ≤ C/|Ω|.
(⇒ Strichartz estimates by Keel-Tao)
Domains with boundary (e.g. Rn+): ... ???
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Literature
After more than 80 years first analytical verification:
Periodic domains (e.g. torus):I Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ.
Math. J., ’03 Russian Math. Surveys
Crucial pre-condition: uniformness in Ω of local results!
Whole space Rn:I J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math.
Appl.I and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ...
Crucial ingredients:I exp(−t(∆ + PΩe3×)) = exp(−t∆) exp(tPΩe3×),
I dispersive estimates:
∣∣∣∣∫R3
exp(ix · ξ + iΩξ3/|ξ|)ψ(ξ) dξ
∣∣∣∣ ≤ C/|Ω|.
(⇒ Strichartz estimates by Keel-Tao)
Domains with boundary (e.g. Rn+): ... ???
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21
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Literature
After more than 80 years first analytical verification:
Periodic domains (e.g. torus):I Babin, Mahalov, Nicolaenko ’97, Asymptotic Anal., ’99 + ’01, Indiana Univ.
Math. J., ’03 Russian Math. Surveys
Crucial pre-condition: uniformness in Ω of local results!
Whole space Rn:I J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math.
Appl.I and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ...
Crucial ingredients:I exp(−t(∆ + PΩe3×)) = exp(−t∆) exp(tPΩe3×),
I dispersive estimates:
∣∣∣∣∫R3
exp(ix · ξ + iΩξ3/|ξ|)ψ(ξ) dξ
∣∣∣∣ ≤ C/|Ω|.
(⇒ Strichartz estimates by Keel-Tao)
Domains with boundary (e.g. Rn+): ... ???
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21
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Contents
1 Rotating fluids
2 The Taylor-Proudman theorem
3 The Ekman boundary layer problem
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Transformed equations and aims
The problem:
(EBLP)
∂tv − ν∆v + (v ,∇)v + Ωe3 × v + (UE · ∇)v + v 3∂3UE = −∇p
div v = 0v |∂G = 0v |t=0 = v0
considered in G = (R2 × (0, d))× (0,T ) for d ∈ [δ,∞].
Main requirements:
Prove results on well-posedness and stability in a functional setting such that
(1) nondecaying, like almost periodic, perturbations of UE are included;
(2) results are uniform in Ω.
Note: This rules out an approach in all standard function spaces such as:
Lp, B0∞,1(Lp), L∞, BUC , Cα, ...
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Functional analytic setting
Idea for ground space
FM0(R2, L2(R+)3) =
v : v L2(R+)3-valued Radon measure, v(0) = 0
(Diestel, Uhl ’77: Vector Measures; important: Radon-Nikodym property)
Motivation 1: FM0(L2) 3 v0(x) :=∑∞
j=1 ajeiλj ·x , x ∈ R3, λj 6= 0
Motivation 2: - Giga, Inui, Mahalov, Matsui, local ex. in FM0(R3),Hokkaido Math. J. ’06, uniform in Ω ,
- Giga, Inui, Mahalov, S., global ex. in FM`(R3) Adv. Differ. Equ ’07;in FM−1
0 (R3) Indiana Univ. Math. J. ’08, uniform in Ω .
Motivation 3:
Lemma (operator-valued multiplier result)
‖F−1σF‖L (FM(L2)) = ‖σ‖L∞(R2\0,L (L2(R+)3)) = ‖F−1σF‖L (L2(R3+)3)).
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Functional analytic setting
Idea for ground space
FM0(R2, L2(R+)3) =
v : v L2(R+)3-valued Radon measure, v(0) = 0
(Diestel, Uhl ’77: Vector Measures; important: Radon-Nikodym property)
Motivation 1: FM0(L2) 3 v0(x) :=∑∞
j=1 ajeiλj ·x , x ∈ R3, λj 6= 0
Motivation 2: - Giga, Inui, Mahalov, Matsui, local ex. in FM0(R3),Hokkaido Math. J. ’06, uniform in Ω ,
- Giga, Inui, Mahalov, S., global ex. in FM`(R3) Adv. Differ. Equ ’07;in FM−1
0 (R3) Indiana Univ. Math. J. ’08, uniform in Ω .
Motivation 3:
Lemma (operator-valued multiplier result)
‖F−1σF‖L (FM(L2)) = ‖σ‖L∞(R2\0,L (L2(R+)3)) = ‖F−1σF‖L (L2(R3+)3)).
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Functional analytic setting
Idea for ground space
FM0(R2, L2(R+)3) =
v : v L2(R+)3-valued Radon measure, v(0) = 0
(Diestel, Uhl ’77: Vector Measures; important: Radon-Nikodym property)
Motivation 1: FM0(L2) 3 v0(x) :=∑∞
j=1 ajeiλj ·x , x ∈ R3, λj 6= 0
Motivation 2: - Giga, Inui, Mahalov, Matsui, local ex. in FM0(R3),Hokkaido Math. J. ’06, uniform in Ω ,
- Giga, Inui, Mahalov, S., global ex. in FM`(R3) Adv. Differ. Equ ’07;in FM−1
0 (R3) Indiana Univ. Math. J. ’08, uniform in Ω .
Motivation 3:
Lemma (operator-valued multiplier result)
‖F−1σF‖L (FM(L2)) = ‖σ‖L∞(R2\0,L (L2(R+)3)) = ‖F−1σF‖L (L2(R3+)3)).
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21
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Functional analytic setting
Idea for ground space
FM0(R2, L2(R+)3) =
v : v L2(R+)3-valued Radon measure, v(0) = 0
(Diestel, Uhl ’77: Vector Measures; important: Radon-Nikodym property)
Motivation 1: FM0(L2) 3 v0(x) :=∑∞
j=1 ajeiλj ·x , x ∈ R3, λj 6= 0
Motivation 2: - Giga, Inui, Mahalov, Matsui, local ex. in FM0(R3),Hokkaido Math. J. ’06, uniform in Ω ,
- Giga, Inui, Mahalov, S., global ex. in FM`(R3) Adv. Differ. Equ ’07;in FM−1
0 (R3) Indiana Univ. Math. J. ’08, uniform in Ω .
Motivation 3:
Lemma (operator-valued multiplier result)
‖F−1σF‖L (FM(L2)) = ‖σ‖L∞(R2\0,L (L2(R+)3)) = ‖F−1σF‖L (L2(R3+)3)).
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21
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Well-posedness and (in-) stability
Theorem (Giga, S. ’13, J. Math. Fluid Mech., ’15 Ark. Mat.)
Let d ∈ (δ,∞), U∞δ/µ < 1/√
2 and set κ = 2(µ−√
2U∞δ).
Linear:
‖ exp(−tASCE )‖L (FM(L2)) ≤ exp(−2κ/d2) (t ≥ 0)
‖∇ exp(−tASCE )v0‖L2((0,d),FM(L2)) ≤ ‖v0‖FM(L2)/√
2κ.
Nonlinear: If ‖u0 −UE‖FM(L2) < πκ/3 · 21/4√
d we have
‖u(t)−UE‖FM(L2) ≤ 2 exp(−2κt)‖u0 −UE‖FM(L2) (t ≥ 0).
Note: Estimates are uniform in Ω.
Theorem (Fischer, S. ’13, Discr. Cont. Dyn. Sys.)
If U∞δ/µ > 55, then the Ekman spiral UE is unstable in FM0(R2, L2(0, d)) and inL2(R2 × (0, d)).
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 18 / 21
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Well-posedness and (in-) stability
Theorem (Giga, S. ’13, J. Math. Fluid Mech., ’15 Ark. Mat.)
Let d ∈ (δ,∞), U∞δ/µ < 1/√
2 and set κ = 2(µ−√
2U∞δ).
Linear:
‖ exp(−tASCE )‖L (FM(L2)) ≤ exp(−2κ/d2) (t ≥ 0)
‖∇ exp(−tASCE )v0‖L2((0,d),FM(L2)) ≤ ‖v0‖FM(L2)/√
2κ.
Nonlinear: If ‖u0 −UE‖FM(L2) < πκ/3 · 21/4√
d we have
‖u(t)−UE‖FM(L2) ≤ 2 exp(−2κt)‖u0 −UE‖FM(L2) (t ≥ 0).
Note: Estimates are uniform in Ω.
Theorem (Fischer, S. ’13, Discr. Cont. Dyn. Sys.)
If U∞δ/µ > 55, then the Ekman spiral UE is unstable in FM0(R2, L2(0, d)) and inL2(R2 × (0, d)).
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 18 / 21
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Operator-valued dispersive estimates
Wanted:
∫Rn
e iλϕ(ξ)ψ(ξ) dξ ∼ 1/λ
for ϕ ∈ C∞(suppψ,L (H)), ϕ = ϕsym + ϕas/Ω, ϕ(ξ)ϕ(η) = ϕ(η)ϕ(ξ) (A)(For Ekman: H = L2(0, d)3, λ = Ωt, ϕ(ξ) =
(x · ξ/t − iFASCEF−1
)/Ω
)
Lemma
supξ‖(∇ϕ(ξ)T∇ϕ(ξ))−1‖L (H) ≤ C ⇒ ‖
∫Rn
e iλϕ(ξ)ψ(ξ) dξ‖L (H) ≤ C/λ (λ > 0).
Pf: ∇e iλϕ(ξ) = iλe iλϕ(ξ)∇ϕ(ξ) .... .
Cor: holds for Ekman in case that |x | > κΩt.
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 19 / 21
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Operator-valued dispersive estimates
Wanted:
∫Rn
e iλϕ(ξ)ψ(ξ) dξ ∼ 1/λ
for ϕ ∈ C∞(suppψ,L (H)), ϕ = ϕsym + ϕas/Ω, ϕ(ξ)ϕ(η) = ϕ(η)ϕ(ξ) (A)(For Ekman: H = L2(0, d)3, λ = Ωt, ϕ(ξ) =
(x · ξ/t − iFASCEF−1
)/Ω
)Lemma
supξ‖(∇ϕ(ξ)T∇ϕ(ξ))−1‖L (H) ≤ C ⇒ ‖
∫Rn
e iλϕ(ξ)ψ(ξ) dξ‖L (H) ≤ C/λ (λ > 0).
Pf: ∇e iλϕ(ξ) = iλe iλϕ(ξ)∇ϕ(ξ) .... .
Cor: holds for Ekman in case that |x | > κΩt.
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 19 / 21
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Operator-valued dispersive estimates
Case |x | ≤ κΩt:
Lemma
Let ϕ ∈ C∞(suppψ,L (H)), ϕ = ϕsym + ϕas/Ω, ϕ(ξ)ϕ(η) = ϕ(η)ϕ(ξ). Then
supξ,η‖[ξT∇2ϕsym(η)ξ
]−1 ‖L (H) ≤ C ⇒ ‖∫Rn
e iλϕ(ξ)ψ(ξ) dξ‖L (H) ≤ C/λ (λ > 0).
Pf:
‖∫Rn
e iλϕ(ξ)ψ(ξ) dξv‖2H =
⟨∫Rn
∫Rn
e iλ[ϕ(ξ+η)−ϕ(η)∗
]ψ(ξ + η)ψ(η) dξ dη v , v
⟩
∇[ϕ(ξ + η)− ϕ(η)∗
]= ∇ϕsym(ξ + η)−∇ϕsym(η) +
1
Ω
[∇ϕas(ξ + η) +∇ϕas(η)
]∼ ∇2ϕsym(hξ + η)ξ +
1
Ω
[∇ϕas(ξ + η) +∇ϕas(η)
].
For Ekman: generalize (A) suitably .... work in progress ....
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 20 / 21
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Happy Birthday Yoshi !!!
... and thank you for 12 years ofmentor-, colleage-, and friendship!
Jurgen Saal (HHU Dusseldorf) Fluid flow and rotation Darmstadt 19.6.2015 21 / 21