enhanced dissipation and transition threshold for the 3-d ... · shijin ding (scnu) 3-d poiseuille...

Enhanced dissipation and Transition threshold for the3-D Poiseuille flow in a channel

Shijin Ding (¶?)

School of Mathematical SciencesSouth China Normal University

Joint work withZhilin Lin () and Zhifei Zhang (Ù)

December, 2019, NUS

Shijin Ding (SCNU) 3-D Poiseuille flow December, 2019, NUS 1 / 41

Beginning with Reynolds experiment

The Reynolds number:

Re :=ρUL

ν,

ρ : density; U : velocity; L : diameter of pipe; ν : viscosity.


Phenomenon and results of the Reynolds experiment

Re small: laminar flow ←→ (a);

Re: transition between laminar flow and turbulence ←→ (b);

Re large: turbulence ←→ (c).


The Navier-Stokes equations

The classical model in (x, y) ∈ R2 and (x, y, z) ∈ R3: ∂tv − ν∆v + (v · ∇)v +∇q = 0,∇ · v = 0,v|t=0 = v0,

(1.1)

where

ν ∼ Re−1 > 0: viscosity;

v ∈ Rn(n = 2, 3): velocity fields;

q: pressure;

Re large←→ random motion of the molecules←→turbulence.


The Euler equations

The Euler equations in (x, y) ∈ R2 and (x, y, z) ∈ R3(ν = 0 in (1.1)): ∂tv + (v · ∇)v +∇q = 0,∇ · v = 0,v|t=0 = v0,

(1.2)

where

v ∈ Rn(n = 2, 3): velocity fields;

q: pressure.


The shear flows

The shear flows:Us = U(y)e1

for some U(y), are both the solution to N-S and Euler.Some important shear flows in physics:

The plane Couette flow: U(y) = y;

The plane Poiseuille flow: U(y) = 1− y2;

The pipe Poiseuille flow (Hagen-Poiseuille flow): (1− r2, 0, 0);

The Kolmogorov flow: U(y) = sin ay or cos ay for some constanta ∈ R.


Formulation of stability problem around the shear flows

Let u = v − Us, then the perturbed equations around Us read as ∂tu− ν∆u+ Us · ∇u+ u · ∇Us +∇P = −u · ∇u,∇ · u = 0,u|t=0 = u0.

(1.3)

The linearized problem can be rewritten as ∂tu+ L (t)u = 0,∇ · u = 0,u|t=0 = u0,

(1.4)

whereL (t)u := −ν∆u+ Us · ∇u+ u · ∇Us +∇P.


The classical analysis

Some classical concepts to study the stability problem:

Spectral stability and instability: σ(L) ∩ c ∈ C : Re c > 0 = ∅;σ(L) ∩ c ∈ C : Re c > 0 6= ∅; σ(L) ∩ c ∈ C : Re c = 0 6= ∅;Lyapunov (nonlinear) stability: the equilibrium uE is called stableif for the perturbation equation: For any ε > 0, there exists δ > 0such that if ‖uin‖X < δ, then ‖u(t)‖Y < ε for all t > 0;

All the above analysis are considered for fixed viscosity (or Reynoldsnumber)!

However, as Reynolds experiment shows that many laminar flowsare very very sensitive to Reynolds number, which maybe stablefor fixed Reynolds number, but spontaneously transit into turbu-lence as Reynolds number tending to infinity!!! (How to define thecritical Reynolds number for every laminar flow?)


The Sommerfeld paradox

For example, consider the Couette flow, the classical spectrum anal-ysis yields that it is stable for any fixed viscosity (or Reynolds number)(V. A. Romanov, 1973).

However, in the experiment, Couette flow would be unstable undersmall perturbation at high Reynolds number!

This leads to a contradiction, i.e., Sommerfeld paradox!


The Sommerfeld paradox

Figure: Experiment result about the Couette flow (Trefethen et al. Science, 1993)


Examples

To understand the relationship among these stability concepts, weconsider

∂t

(x1

x2

)=

(0 −10 0

)(x1

x2

), (1.5)

The matrix

(0 −10 0

)is not diagonalizable and has spectrum reduced

to 0, therefore it is neutrally stable, but it is not Lyaunov stable:x1(t) = x1(0)− tx2(0),x2(t) = x2(0),

(1.6)

Grow with t !


Examples

Above example can be modified to make it spectrally stable andLyapunov stable:

∂t

(x1

x2

)=

(−ν −10 −ν

)(x1

x2

), (1.7)

The solutions are spectrally stable and Lyapunov linearly stable. How-ever, as ν → 0, the system degenerates, and

supt‖(x1, x2)(t)‖ . ν−1‖(x1, x2)(0)‖.

The system is linearly stable for any fixed ν > 0, but the solutions aregrowing very large as ν → 0 (very sensitive to Reynolds number!)


The stability transition problem

To understand the paradox and explain our problem, we introducethe quantitative asymptotic stability (instability).

Definition (Quantitative asymptotic stability)

Given two norms X and Y , the equilibrium Us is called quantitativestable (from X to Y ) with exponent γ if

‖u0‖X << νγ

=⇒ ‖u(t)‖Y << 1,∀t ≥ 0, and ‖u(t)‖Y → 0 as t→∞.

Qualification of δ in the definition of Lyapunov stability byνγ !!! (where γ depends on the space X.)


The stability transition problem

A fundamental problem (Trefethen et al. Science, 1993; Masmoudiet al. 2015):

Given a norm ‖ · ‖X , determine a γ = γ(X) such that

‖u0‖X ≤ νγ =⇒ stability,

‖u0‖X νγ =⇒ instability.

γ: the transition threshold in the applied literature.

Our goal:

Answer part of the above problem for the 3-D Poiseuille flow.


Formulation of the problem

The Navier-Stokes equations around the 3-D Poiseuille flow (1−y2, 0, 0)in a channel (x, y, z) ∈ T× I × T(I = (−1, 1)):

∂tu− ν∆u+ (u · ∇)u+ (1− y2)∂xu+

−2yu2

00

+∇P = 0,

∇ · u = 0,u|t=0 = u0(x, y, z).

(1.8)

The Navier-slip boundary conditions:

u · n = 0, ω × n = 0 on y = ±1, (1.9)

ω = ∇× u: vorticity; n: unit outer normal on the boundaries.


Important formulation of our problem

The important formulation in our problem:(∂t − ν∆ + (1− y2)∂x)∆u2 + 2∂xu2 = −∆(u · ∇u2)− ∂y(∆p),(∂t − ν∆ + (1− y2)∂x)ω2 − 2y∂zu2 = ∇ · (ωu2 − uω2),(∆u2, ω2)(t;x,±1, z) = (0, 0),(∆u2, ω2)|t=0 = (∆u2(0), ω2(0)),

(1.10)

ω2 = ∂zu1 − ∂xu3; p = −∆−1

(3∑

i,j=1∂iuj∂jui

).


The linearized problem

The linearized Navier-Stokes equations:(∂t + L1)∆u2 = 0,(∂t + L2)ω2 = 2y∂zu2,(∆u2, ω2)(t;x,±1, z) = (0, 0),(∆u2, ω2)|t=0 = (∆u2(0), ω2(0)).

(1.11)

The linearized operators:

L1 = −ν∆ + (1− y2)∂x + 2∂x∆−1 (with nonlocal term)(1.12)

andL2 = −ν∆ + (1− y2)∂x, (1.13)


Some known results (Couette flow)

When Ω = T× R× T,

if the perturbation is in Gevrey class, then γ = 1 (Bedrossian etal., arXiv, 2015);

if the perturbation is in Sobolev space Hσ(σ > 9/2), then γ ≤ 32

(Bedrossian et al., Ann. Math.,2017);

if the perturbation is in Sobolev space H2, then γ ≤ 1 (Wei,Zhang, arXiv, 2018).


Some known results (Couette flow)

When Ω = T× R,if the perturbation is in Gevrey class, then γ = 0 (Bedrossian etal., ARMA, 2016);

if the perturbation is in Sobolev space Hs(s > 1), then γ ≤ 12

(Bedrossian et al., J. Nonlinear Sci., 2018);

if the perturbation is in Sobolev space Hσ(σ ≥ 40), then γ ≤ 13

(Masmoudi, Zhao, arXiv, 2019)

When Ω = T× I(I = (−1, 1)),

if the perturbation is in Sobolev space H2, then γ ≤ 12

(Chen, Li,Wei, Zhang, arXiv, 2018)


Some known results (other flows)

Kolomogorov flow:

if the perturbation is in Sobolev space H2 and Ω = T2, thenγ ≤ 2

3+ (Wei, Zhao, Zhang, arXiv, 2017);

if the perturbation is in Sobolev space H2 and Ω = T3, then γ ≤ 74

(Li, Wei, Zhang, CPAM, 2019)

Poiseuille flow:

if the perturbation is in Sobolev space and Ω = T × R, thenγ ≤ 3

4+ (Zelati et al., arXiv, 2019)


Linear effects

The main linear effects:

3-D lift-up effect;

Long-wave effect: (x, z) ∈ R × T (the main reason resulting ininstability at high Reynolds number);

Boundary layer effect: ν → 0;

Inviscid damping;

Enhanced dissipation.


3-D lift-up effect (Couette flow)

To express the effect, as an example, we consider the linearized Navier-Stokes around Couette flow (y, 0, 0):

∂tu− ν∆u+ y∂xu+ (u2, 0, 0) +∇P = 0.

The zero mode u(t; y, z) =∫T udx enjoys

∂tu− ν∆u+ (u2, 0, 0) = 0.

It is easy to see that

u1 ∼ te−νt ∼ t for t . ν−1.

This is so called the transient growth.


Inviscid damping (Couette flow)

The linearized 2-D Euler equation around Couette flow (y, 0):

∂tω + y∂xω = 0 =⇒ ω(t;x, y) = ω0(x− ty, y).

The result of Orr (1907): if∫T ω0dx = 0, then

‖u(t)‖L2 ∼ t−1 → 0 as t→∞.

This is so called the inviscid damping.


Enhanced dissipation (Couette flow)

The linearized 2-D Navier-Stokes equation around Couette flow inthe vorticity:

∂tω − ν∆ω + y∂xω = 0, ω|t=0 = ω0.

If∫T ω0dx = 0, then

‖ω(t)‖L2 . e−ν13 t‖ω0‖L2 .

This decay rate ν13 is much faster than the heat diffusion rate ν, which

is so called the enhanced dissipation.


Our main results (Linear enhanced dissipation)

Theorem (D.-Lin-Zhang, 2019)

For any 0 < ν ≤ 1, and k = (k1, k3) ∈ Z2 with k1 6= 0, the solution of(1.11) satisfies the following estimates

‖Φ(t; k1, ·, k3)‖L2 ≤ Ce−at‖Φ(0; k1, ·, k3)‖L2 ,

and

‖ω2(t; k1, ·, k3)‖L2 ≤Ce−at‖ω2(0; k1, ·, k3)‖L2

+Ce−at(1 + at)

|k1|‖Φ(0; k1, ·, k3)‖L2 ,

where a = c|k1ν|12 + ν|k|2 with c ∈ (0, 1), Φ := ∆u2; φ(k1, y, k3):

Fourier transform φ(x, y, z) with respect to the directions (x, z).


Our main results (Stability threshold of Poiseuille flow(1− y2, 0, 0))

Theorem (D.-Lin-Zhang, 2019)

Suppose that there exists c0 ∈ (0, 1) such that ‖u0‖H2 +‖∂zu0‖H2 ≤ c0ν74 ,

then the solution u of (1.10) is global in time with

‖∂zu2‖H2 + ‖u‖H2 + ec′√νt‖(∆u2)6=‖L2

+ ec′√νt‖∂z(∆u2)6=‖L2 + ec

′√νt‖∂xω2‖L2

+ ‖P0u3‖H1 + ‖∂zP0u3‖H1 ≤ C(‖u0‖H2 + ‖∂zu0‖H2),

and‖u‖H2 + ‖∂zu‖H2 ≤ Cν−1(‖u0‖H2 + ‖∂zu0‖H2),

where

P0f =

∫Tf(x, y, z)dx, f 6= = f − P0f.


Some remarks

The decay rate√ν in enhanced dissipation corresponds to the

structure of Poiseuille flow (Couette flow: ν1/3), which is fasterthan that of hear diffussion;

The transition threshold γ ≤ 74

also holds for the 3-D Kolmogorovflow and it may not be optimal;

Compared with the results obtained by Zelati et al. for the 2-DPoiseuille flow in T×R, there is no logarithmic loss in our results.


Main idea of our proof

Resolvent estimates (sharp): including the resolvent estimates forOrr-Sommerfeld equations with and without the nonlocal term;

Pseudospectra bounds and semigroup bound (sharp): obtainedvia Gearhart-Pruss type lemma (Wei, 2019, SCM)=⇒ enhanceddissipation

Continuity argument (for nonlinear stability):Suppose that there exists small ε1 ∈ (0, 1) (independent of ν, k, λ),such that for any 0 < ν < 1, T > 0, if the solution ∂kzu ∈C([0, T ];H2) ∩ L2([0, T ;H3)(k = 0, 1) of (1.10) satisfies

E0(T ) ≤ ε1ν, E1(T ) ≤ ε1ν34 (needed in estimates),

=⇒ E0(T ) ≤ C1(‖u0‖H2 + ‖∂zu0‖H2),

E1(T ) ≤ C1ν−1(‖u0‖H2 + ‖∂zu0‖H2),


Main idea of our proof

where C1 is independent of ν, k, λ and

E0(T ) = supt∈[0,T ]

(‖∂zu2‖H2 + ‖u‖H2 + ec

′√νt‖(∆u2) 6=‖L2

+ ec′√νt‖∂z(∆u2) 6=‖L2 + ec

′√νt‖∂xω2‖L2

+ ‖P0u3‖H1 + ‖∂zP0u3‖H1

),

E1(T ) = supt∈[0,T ]

(‖u‖H2 + ‖∂zu‖H2

).

Continuity argument: Let (‖u0‖H2 + ‖∂zu0‖H2) ≤ c0ν7/4, E0(T ) ≤ ε1ν,

E1(T ) ≤ ε1ν3/4. Then choosing c0 = ε1

2C1yields E0(T ) ≤ ε1

2 ν and

E1(T ) ≤ ε12 ν

3/4 from above estimates!!! (This is the continuity argument!)


The Orr-Sommerfeld equations with Navier-slip boundaryconditions

The Orr-Sommerfeld equations without the nonlocal term:−ν(∂2

y − |k|2)w + ik1(1− y2 − λ)w = F1,w(±1) = 0.

(1.14)

The Orr-Sommerfeld equations with the nonlocal term:−ν(∂2

y − |k|2)w + ik1[(1− y2 − λ)w + 2ϕ] = F,w(±1) = 0,

(1.15)

where (∂2y − |k|2)ϕ = w with ϕ(±1) = 0, the nonlocal term:

ϕ = (∂2y − |k|2)−1w.


Sketch of the proof

Step 1: Resolvent estimates.

Lemma (Without the nonlocal term)

Suppose that w ∈ H2(I) is the solution of (1.14) with λ ∈ R andF1 ∈ L2(I). Then there holds that

ν12 |k1|

12‖w‖L2 ≤ C‖F1‖L2 . (1.16)

Lemma (With the nonlocal term)

Suppose that w ∈ H2(I) is the solution of (1.15) with λ ∈ R andF ∈ L2(I). Then there holds that

ν12 |k1|

12‖w‖L2 ≤ C‖F‖L2 . (1.17)


Sketch of the proof: Resolvent estimates

The proofs for the above two Lemmas are very lengthy (19 pages!)since the Poiseuille flow is a non-monotone flow. Especially for thecase with nonlocal term, the proof is much more complicated. Anexample without nonlocal term:

For δ ∼ ν14 |k1|−

14 and λ > 1, key estimate:∫ 1

−1

(λ− 1 + y2)|w|2dy ≥∫

[−1,1]\(−δ,δ)(λ− 1 + y2)|w|2dy

& δ2

∫[−1,1]\(−δ,δ)

|w|2dy,

Other case can be obtained by similar arguments.


Sketch of the proof: Semigroup bound

Step 2: Pseudospectra bounds, semigroup bound and enhanced dissi-pation.

Recall

L1 = −ν(∂2y − |k|2) + ik1[(1− y2) + 2(∂2

y − |k|2)−1]

andL2 = −ν(∂2

y − |k|2) + ik1(1− y2).

Introduce

Ψ(A) := inf‖(A− iλ)‖ : f ∈ D(A), λ ∈ R, ‖f‖ = 1.

Lemma (Wei, SCM, 2019)

Let A be an m-accretive operator on a Hilbert spaces X, then ‖e−tA‖ ≤e−tΨ(A)+π

2 for any t ≥ 0.


Sketch of the proof: Enhanced dissipation

Lemma (Semigroup bound)

There exist constants C, c > 0, independent of ν such that

‖e−tLig6=‖L2 ≤ Ce−cν12 t−νt‖g6=‖L2 , i = 1, 2. (1.18)

=⇒ ‖Φ(t; k1, ·, k3)‖L2 ≤ Ce−cν12 t−νt‖Φ(0; k1, ·, k3)‖L2 .

This is the enhanced dissipation for (∆u2) 6=. (follows from the semi-group bound directly)


Sketch of the proof: Enhanced dissipation

To study the enhanced dissipation for g, we consider (∂t + L1)φ = 0,

(∂t + L2)g = 2ik3y(∂2y − |k|2)−1φ,

(φ, g)|t=0 = (φ0, g0).

(1.19)

Idea (deal with the nonlocal term in (1.19)2):

g1 = g +k3

k1

yφ,=⇒ (∂t + L2)g1 = −ν k3

k1

y∂yφ. =⇒ OK!

The enhanced dissipation can be obtained by semigroup bound.


Sketch of the proof: Nonlinear problem

Step 3: Nonlinear stability (Space-time estimates and closing the en-ergy estimates).

We consider the nonlinear problem.(∂t + L1)∆u2 = ∇ · f,(∂t + L2)ω2 − 2y∂zu2 = ∇ ·G,∆u2(t;x,±1, z) = ω2(t;x,±1, z) = 0,(∆u2, ω2)|t=0 = (∆u2(0), ω2(0)),

(1.20)

where f = −∇(u · ∇u2) − (0,∆p, 0) and G = (ω1, ω2, ω3)u2 −(u1, u2, u3)ω2.



We establish the following estimates.

Lemma

‖∂kz (∆u2) 6=‖2Xc′

. ‖∂kz (∆u2) 6=(0)‖2L2 + ν−1‖ec′

√νt∂kz f 6=‖2

L2L2 ,

(1.21)and

‖∂xω2‖2Xc′

. ‖∂xω2(0)‖2L2 + ‖(∆u2)6=(0)‖2

L2 + ‖∂z(∆u2)6=(0)‖2L2

+ ν−1(‖ec′

√νt∂xG‖2

L2L2 + ‖ec′√νtf 6=‖2

L2L2 + ‖ec′√νt∂zf 6=‖2

L2L2

),

(1.22)where k = 0, 1.



The energy functional:

‖u‖2Xc′

= ‖ec′√νtu‖2

L∞L2 +√ν‖ec′

√νtu‖2

L2L2 + ν‖ec′√νt∇u‖2

L2L2 .

In the norm ‖ · ‖Xc′ , the first two parts correspond to the enhanced

dissipation whose decay rate e−c√νt is resulted from the sharp resolvent

estimates, and the last part is due to the combined effect of heatdiffusion and enhanced dissipation.



For the nonlinear problem, the enhanced dissipation with rate√ν plays

an important role in nonlinear problem.For example:

‖ec′√νt∂xG‖2

L2L2 + ‖ec′√νtf 6=‖2

L2L2 + ‖ec′√νt∂zf 6=‖2

L2L2

.E20ν−1‖∂xω2‖2

Xc′+ E2

0ν−1‖∆u2‖2

Y0

+ E21ν− 1

2 (‖(∆u2)6=‖2Xc′

+ ‖∂z(∆u2) 6=‖2Xc′

+ ‖∂xω2‖2Xc′

).

and

‖(∆u2)6=‖2Xc′

+ ‖∂z(∆u2)6=‖2Xc′

+ ‖∂xω2‖2Xc′

.(‖u(0)‖H2 + ‖∂zu(0)‖H2)2 + ν−1(E2

0ν−1‖∂xω2‖2

Xc′+ E2

0ν−1‖∆u2‖2

Y0

+ E21ν− 1

2 (‖(∆u2)6=‖2Xc′

+ ‖∂z(∆u2)6=‖2Xc′

+ ‖∂xω2‖2Xc′

)).



To close the energy estimates, the following estimates are needed:

1∑k=0

(‖∂kzP0∆u2‖2

Y0+ ‖∂kz∇P0u3‖2

Y0+ ν−1‖∂kz∆p‖2

L2L2

+ ‖∂kz (∆u2)6=‖2Xc′

)+ ‖∂xω2‖2

Xc′≤ C(‖u(0)‖H2 + ‖∂zu(0)‖H2)2,

∑k=0,1

(‖∂kz∆P0u3‖2Y0

+ ‖∂kz∆u6=‖2Xc′

) . ν−32 (‖u0‖H2 + ‖∂zu0‖H2)2,

E0 ≤ C(‖u0‖H2 + ‖∂zu0‖H2), E1 ≤ Cν−1(‖u0‖H2 + ‖∂zu0‖H2).

Finally, the standard continuity argument yields the conclusion.


Thanks for your attention!


enhanced dissipation and transition threshold for the 3-d ... · shijin ding (scnu) 3-d poiseuille...

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