Fluid flow and rotation: a fascinating interplay J urgen Saal - - PowerPoint PPT Presentation

Fluid flow and rotation: a fascinating interplay

J¨ urgen Saal Mathematics for Nonlinear Phenomena: Analysis and Computation International Conference in honor of

Professor Yoshikazu Giga

• n his 60th birthday

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 1 / 21

About 12 years ago in Sapporo there was ...

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 2 / 21

... some years later ...

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 3 / 21

Contents

1

Rotating fluids

2

The Taylor-Proudman theorem

3

The Ekman boundary layer problem

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 4 / 21

Contents

1

Rotating fluids

2

The Taylor-Proudman theorem

3

The Ekman boundary layer problem

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 5 / 21

The Ekman boundary layer problem (EBLP)

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

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 6 / 21

Modeling of the problem

Navier-Stokes equations with Coriolis term (EBLP)        ∂tv − ν∆v + (v, ∇)v + Ωe3 × v = −∇p div v = v|∂G = UE|∂G v|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 ”.

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 7 / 21

Literature

Geophysical: 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 boundary layers 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.

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 8 / 21

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, M¨ uller, Shibata, Neustupa, Necasova, Hieber, Geißert, Heck, Dintelmann, Penel, Disser, Maekawa, ...

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 9 / 21

Rotating obstacle vs Ekman

’Rotating obstacle’ situation (e.g. propeller) additional forces: Ωe3 × u Corolis + Ω2 2 e3 × (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 2 e3 × (e3 × ξ)

• centrifugal

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 10 / 21

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)

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

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:

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

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

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 11 / 21

Contents

1

Rotating fluids

2

The Taylor-Proudman theorem

3

The Ekman boundary layer problem

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 12 / 21

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 no variation of the velocity field in the direction parallel to the axis of rotation.

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

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 no variation 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 dz u = e3 · ∇u ≈ 0 ⇒ u(x, y, z) ≈ u(x, y)

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

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 no variation 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 dz u = e3 · ∇u ≈ 0 ⇒ u(x, y, z) ≈ u(x, y) ⇒ flow is two-dimensional ⇒ global well-posedness !

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 13 / 21

Literature

After more than 80 years first analytical verification: Periodic domains (e.g. torus):

◮ 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!

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Literature

After more than 80 years first analytical verification: Periodic domains (e.g. torus):

◮ 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:

◮ J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math.

Appl.

◮ and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ...

Crucial ingredients:

◮ exp(−t(∆ + PΩe3×)) = exp(−t∆) exp(tPΩe3×), ◮ dispersive estimates:

• R3 exp(ix · ξ + iΩξ3/|ξ|)ψ(ξ) dξ
• ≤ C/|Ω|.

(⇒ Strichartz estimates by Keel-Tao)

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Literature

After more than 80 years first analytical verification: Periodic domains (e.g. torus):

◮ 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:

◮ J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier ’02, Stud. Math.

Appl.

◮ and Koba, Mahalov, Yoneda, Konieczny, Koh, Lee, Takada, ...

Crucial ingredients:

◮ exp(−t(∆ + PΩe3×)) = exp(−t∆) exp(tPΩe3×), ◮ dispersive estimates:

• R3 exp(ix · ξ + iΩξ3/|ξ|)ψ(ξ) dξ
• ≤ C/|Ω|.

(⇒ Strichartz estimates by Keel-Tao)

Domains with boundary (e.g. Rn

+): ... ???

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 14 / 21

Contents

1

Rotating fluids

2

The Taylor-Proudman theorem

3

The Ekman boundary layer problem

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 15 / 21

Transformed equations and aims

The problem: (EBLP)        ∂tv − ν∆v + (v, ∇)v + Ωe3 × v + (UE · ∇)v + v 3∂3UE = −∇p div v = v|∂G = v|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, C α, ...

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 16 / 21

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-Nikod´ ym property)

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21

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-Nikod´ ym property) Motivation 1: FM0(L2) ∋ v0(x) := ∞

j=1 ajeiλj·x,

x ∈ R3, λj = 0

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21

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-Nikod´ ym property) Motivation 1: FM0(L2) ∋ v0(x) := ∞

j=1 ajeiλj·x,

x ∈ R3, λj = 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 FMℓ(R3) Adv. Differ. Equ ’07;

in FM−1

0 (R3) Indiana Univ. Math. J. ’08, uniform in Ω .

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21

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-Nikod´ ym property) Motivation 1: FM0(L2) ∋ v0(x) := ∞

j=1 ajeiλj·x,

x ∈ R3, λj = 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 FMℓ(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−1σFL (FM(L2)) = σL∞(R2\{0},L (L2(R+)3)) = F−1σFL (L2(R3

+)3)). J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 17 / 21

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)v0L2((0,d),FM(L2)) ≤ v0FM(L2)/ √ 2κ. Nonlinear: If u0 − UEFM(L2) < πκ/3 · 21/4√ d we have u(t) − UEFM(L2) ≤ 2 exp(−2κt)u0 − UEFM(L2) (t ≥ 0). Note: Estimates are uniform in Ω.

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 18 / 21

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)v0L2((0,d),FM(L2)) ≤ v0FM(L2)/ √ 2κ. Nonlinear: If u0 − UEFM(L2) < πκ/3 · 21/4√ d we have u(t) − UEFM(L2) ≤ 2 exp(−2κt)u0 − UEFM(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 in L2(R2 × (0, d)).

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 18 / 21

Operator-valued dispersive estimates

Wanted:

• Rn eiλϕ(ξ)ψ(ξ) dξ ∼ 1/λ

for ϕ ∈ C ∞(supp ψ, 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

• J¨

urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 19 / 21

Operator-valued dispersive estimates

Wanted:

• Rn eiλϕ(ξ)ψ(ξ) dξ ∼ 1/λ

for ϕ ∈ C ∞(supp ψ, 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

sup

ξ

(∇ϕ(ξ)T∇ϕ(ξ))−1L (H) ≤ C ⇒

• Rn eiλϕ(ξ)ψ(ξ) dξL (H) ≤ C/λ (λ > 0).

Pf: ∇eiλϕ(ξ) = iλeiλϕ(ξ)∇ϕ(ξ) .... . Cor: holds for Ekman in case that |x| > κΩt.

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 19 / 21

Operator-valued dispersive estimates

Case |x| ≤ κΩt:

Lemma

Let ϕ ∈ C ∞(supp ψ, L (H)), ϕ = ϕsym + ϕas/Ω, ϕ(ξ)ϕ(η) = ϕ(η)ϕ(ξ). Then sup

ξ,η

• ξT∇2ϕsym(η)ξ

−1 L (H) ≤ C ⇒

• Rn eiλϕ(ξ)ψ(ξ) dξL (H) ≤ C/λ (λ > 0).

Pf:

• Rn eiλϕ(ξ)ψ(ξ) dξv2

H =

• Rn
• Rn eiλ
• 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 ....

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 20 / 21

Happy Birthday Yoshi !!!

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

J¨ urgen Saal (HHU D¨ usseldorf) Fluid flow and rotation Darmstadt 19.6.2015 21 / 21