fluid flow and rotation: a fascinating interplay [-3mm]j j \ layer thickness ". jurgen saal...

$Page 1: Fluid flow and rotation: a fascinating interplay [-3mm]j j \ layer thickness ". Jurgen Saal (HHU Dusseldo rf) Fluid ow and rotation Darmstadt 19.6.2015 7 / 21 Literature Geophysical:$

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

About 12 years ago in Sapporo there was ...

... some years later ...

Contents

1 Rotating fluids

2 The Taylor-Proudman theorem

3 The Ekman boundary layer problem

Contents

1 Rotating fluids

The Ekman boundary layer problem (EBLP)

Important for: daily weather forecast, tornado and hurricane evolution, aviation,pollen distribution, dew, fog, frost forecasts, air pollution, ....

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 ”.

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.

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, ...

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

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

Linearized spectrum

Spectra:

=⇒

Linearized spectrum

Spectra:

=⇒

Contents

1 Rotating fluids

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 !

(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)

(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)

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+): ... ???

Literature

∣∣∣∣∫R3

∣∣∣∣ ≤ C/|Ω|.

Literature

∣∣∣∣∫R3

∣∣∣∣ ≤ C/|Ω|.

Contents

1 Rotating fluids

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, ...

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)).

FM0(R2, L2(R+)3) =

j=1 ajeiλj ·x , x ∈ R3, λj 6= 0

Motivation 3:

FM0(R2, L2(R+)3) =

j=1 ajeiλj ·x , x ∈ R3, λj 6= 0

Motivation 3:

FM0(R2, L2(R+)3) =

j=1 ajeiλj ·x , x ∈ R3, λj 6= 0

Motivation 3:

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)).

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)).

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.

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.

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 ....

Happy Birthday Yoshi !!!

... and thank you for 12 years ofmentor-, colleage-, and friendship!

fluid flow and rotation: a fascinating interplay [-3mm]j j \ layer thickness ". jurgen saal...

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