# Weak turbulence

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```					                            Weak turbulence

Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier

Benjamin Texier (Paris 7)       Weak turbulence                    1 / 36
Weak turbulence

Turbulence

Benjamin Texier (Paris 7)            Weak turbulence   2 / 36
Weak turbulence

Turbulence

dissipation at small scales

source at large scale.

Benjamin Texier (Paris 7)            Weak turbulence                   2 / 36
Weak turbulence

The Kolmogorov-Zakharov equation

The particle density n satisﬁes

∂t n =            n(k1 )n(k2 )n(k3 ) + n(k)n(k2 )n(k3 )

− n(k)n(k1 )n(k3 ) − n(k)n(k1 )n(k2 )
2    2    2
× δ(k + k1 − k2 − k3 )δ(ω 2 + ω3 − ω1 − ω2 )dk1 dk2 dk3

where ω(k) is a linear dispersion relation.

Benjamin Texier (Paris 7)                   Weak turbulence                3 / 36
Weak turbulence

The Kolmogorov-Zakharov equation

The particle density n satisﬁes

∂t n =            n(k1 )n(k2 )n(k3 ) + n(k)n(k2 )n(k3 )

− n(k)n(k1 )n(k3 ) − n(k)n(k1 )n(k2 )
2    2     2
× δ(k + k1 − k2 − k3 )δ(ω 2 + ω3 − ω1 − ω2 )dk1 dk2 dk3
1        1       1        1
=      n(k)n(k1 )n(k2 )n(k3 )      +       −        −
n(k) n(k1 ) n(k2 ) n(k3 )
2    2    2
× δ(k + k1 − k2 − k3 )δ(ω 2 + ω3 − ω1 − ω2 )dk1 dk2 dk3

where ω(k) is a linear dispersion relation.

Benjamin Texier (Paris 7)                   Weak turbulence                3 / 36
Weak turbulence

The Kolmogorov-Zakharov equation

The particle density n satisﬁes

∂t n =            n(k1 )n(k2 )n(k3 ) + n(k)n(k2 )n(k3 )

− n(k)n(k1 )n(k3 ) − n(k)n(k1 )n(k2 )
2    2     2
× δ(k + k1 − k2 − k3 )δ(ω 2 + ω3 − ω1 − ω2 )dk1 dk2 dk3
1        1       1        1
=      n(k)n(k1 )n(k2 )n(k3 )      +       −        −
n(k) n(k1 ) n(k2 ) n(k3 )
2    2    2
× δ(k + k1 − k2 − k3 )δ(ω 2 + ω3 − ω1 − ω2 )dk1 dk2 dk3

where ω(k) is a linear dispersion relation.

= particle (and energy) ﬂux given by cubic “collision” integral

Benjamin Texier (Paris 7)                   Weak turbulence                3 / 36
Weak turbulence

Why the KZ equation might be relevant

it is irreversible (H theorem);

n(k) ≡ k −β ,             β > 0,

so that
n(λk) ≡ λ−β n(k).
β
Compare with Maxwellians: n(k) ≡ e −k , for which n(λk)       n(k).

Benjamin Texier (Paris 7)            Weak turbulence                    4 / 36
Weak turbulence

Formal derivation of the KZ equation
o
Start from, e.g., dissipative nonlinear Schr¨dinger:
1             1
i∂t ψ + ∆ψ + iγψ = ± √ |ψ|2 ψ ,              γ > 0,   x ∈ R,
ε              ε
√
where linear time scale ∼ 1/ε            nonlinear time scale ∼ 1/ ε.

Benjamin Texier (Paris 7)            Weak turbulence                     5 / 36
Weak turbulence

Formal derivation of the KZ equation
o
Start from, e.g., dissipative nonlinear Schr¨dinger:
1             1
i∂t ψ + ∆ψ + iγψ = ± √ |ψ|2 ψ ,                γ > 0,   x ∈ R,
ε              ε
√
where linear time scale ∼ 1/ε              nonlinear time scale ∼ 1/ ε.

ˆ               itk 2
Factorization:    ϕ(t, k) := ψ(t, k) exp           + γt .
ε

Benjamin Texier (Paris 7)              Weak turbulence                     5 / 36
Weak turbulence

Formal derivation of the KZ equation
o
Start from, e.g., dissipative nonlinear Schr¨dinger:
1             1
i∂t ψ + ∆ψ + iγψ = ± √ |ψ|2 ψ ,                     γ > 0,   x ∈ R,
ε              ε
√
where linear time scale ∼ 1/ε                 nonlinear time scale ∼ 1/ ε.

ˆ                 itk 2
Factorization:       ϕ(t, k) := ψ(t, k) exp             + γt .
ε
Equation in ϕ :
√
i ε∂t ϕ =            exp −t iε−1 Ωk1 ,k2 ,k3 ,k + 2γ

× ϕ(k1 )ϕ(k2 )ϕ(k3 ) δk1 +k2 −k−k3 dk1 dk2 dk3 ,
¯
where
Ω(k1 , k2 , k3 , k) := ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k).

Benjamin Texier (Paris 7)                 Weak turbulence                          5 / 36
Weak turbulence

Statistical setting

Initial datum is a random variable on a probability space Ω :
ˆ
ω ∈ Ω → ψ(0, k, ω).

Ensemble averaging:

ϕ :=            ϕ(ω)dP(ω),             P probability measure on Ω.

Benjamin Texier (Paris 7)                 Weak turbulence                        6 / 36
Weak turbulence

Statistical setting

Initial datum is a random variable on a probability space Ω :
ˆ
ω ∈ Ω → ψ(0, k, ω).

Ensemble averaging:

ϕ :=            ϕ(ω)dP(ω),             P probability measure on Ω.

Two-point correlation function:

¯
N(t, k, k ) := ϕ(t, k)ϕ(t, k ) .

The equation in N involves a four-point correlation function.

Benjamin Texier (Paris 7)                 Weak turbulence                        6 / 36
Weak turbulence

Statistical setting

Initial datum is a random variable on a probability space Ω :
ˆ
ω ∈ Ω → ψ(0, k, ω).

Ensemble averaging:

ϕ :=            ϕ(ω)dP(ω),             P probability measure on Ω.

Two-point correlation function:

¯
N(t, k, k ) := ϕ(t, k)ϕ(t, k ) .

The equation in N involves a four-point correlation function.
Hence hierarchy of equations, i.e., inﬁnite system of coupled integro-diﬀerential
equations.

Benjamin Texier (Paris 7)                 Weak turbulence                        6 / 36
Weak turbulence

Equation in the two-point correlation function

√
i ε(N(t, k, k ) − N(0, k, k ))
t
=                exp −t iε−1 Ωk1 k2 k3 k + 2γ
0
× ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k ) δk1 +k2 −k3 −k dt dk1 dk2 dk3
¯     ¯
t
−                exp −t iε−1 Ωk       k3 k1 k2   + 2γ
0
× ϕ(k)ϕ(k3 )ϕ(k1 )ϕ(k2 ) δk
¯     ¯                   +k3 −k1 −k2 dt   dk1 dk2 dk3 .

Benjamin Texier (Paris 7)                    Weak turbulence                                        7 / 36
Weak turbulence

Equation in the two-point correlation function

Integration by parts: N(t, k, k ) − N(0, k, k ) = I4 + I6 ,

√            exp −t iε−1 Ωk1 k2 k3 k + 2γ
I4 =       ε
Ωk1 k2 k3 k − 2iγε
t
× ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k) δk+k3 −k1 −k2 dk1 dk2 dk3
¯     ¯
0
+
t   exp −t iε−1 Ωk1 k2 k3 k + 2γ
I6 = −
0            Ωk1 k2 k3 k − 2iγε
× ∂t ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k ) δk1 +k2 −k3 −k dt dk1 dk2 dk3
¯     ¯
six-point correlation function
+

Benjamin Texier (Paris 7)                    Weak turbulence                           8 / 36
Weak turbulence

Quasi-gaussian approximation

The quasi-gaussian approximation postulates that correlations are
negligible:

¯
ϕ(t, kj )ϕ(t, kj ) =               n(t, kj )          δkj −k      ,   (1)
σ(j)
1≤j,j ≤s                                 1≤j≤s                σ∈Ss

where
¯
n(t, k) := N(t, k, k) = ϕ(t, k)ϕ(t, k) .
The independence property (1) is not propagated by the nonlinear equation.

Benjamin Texier (Paris 7)                    Weak turbulence                                      9 / 36
Weak turbulence

Cancellation

Under the quasi-gaussian approximation:

ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k) = 0
¯     ¯                            ⇒      {k1 , k2 } = {k3 , k}

and
Ωk1 k2 k1 k2 = 0           (trivial resonance).

exp −t iε−1 Ωk1 k2 k3 k + 2γ
` `                       ´´
√
Z
I4 (t, k, k) = 2 εRe
Ωk1 k2 k3 k − 2iγε
˛t
× ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k) δk+k3 −k1 −k2 dk1 dk2 dk3 ˛
¯     ¯
˛
0
≡ 0.

Benjamin Texier (Paris 7)                      Weak turbulence                                10 / 36
Weak turbulence

Cancellation

Under the quasi-gaussian approximation:

ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k) = 0
¯     ¯                            ⇒      {k1 , k2 } = {k3 , k}

and
Ωk1 k2 k1 k2 = 0           (trivial resonance).

exp −t iε−1 Ωk1 k2 k3 k + 2γ
` `                       ´´
√
Z
I4 (t, k, k) = 2 εRe
Ωk1 k2 k3 k − 2iγε
˛t
× ϕ(k1 )ϕ(k2 )ϕ(k3 )ϕ(k) δk+k3 −k1 −k2 dk1 dk2 dk3 ˛
¯     ¯
˛
0
≡ 0.

e
Null condition, transparency (Joly, M´tivier, Rauch).

Benjamin Texier (Paris 7)                      Weak turbulence                                10 / 36
Weak turbulence

Concentration on the resonant set

If f is smooth and decaying at inﬁnity,

f (k)
lim Im                          dk
γ→0+           Ωk1 k2 k3 k − 2iγ
π                        2    2    2
=−                        f          θ   k1 + k2 − k3 dθ.
S2        2    2    2
k1 + k2 − k3

This gives the KZ equation in the limit γ → 0+ .

Benjamin Texier (Paris 7)                Weak turbulence                          11 / 36
Weak turbulence

Issues

Propagation of chaos.
The quasi-gaussian approximation does not hold. Can it be recovered
asymptotically ?

Benjamin Texier (Paris 7)            Weak turbulence                    12 / 36
Weak turbulence

Issues

Propagation of chaos.
The quasi-gaussian approximation does not hold. Can it be recovered
asymptotically ?
Spatial dependence.
o
We expect the limit for the autonomous Schr¨dinger equation to be a mean-ﬁeld
model (resonant waves not localized in space, interact at all times), rather than a
collisional model such as KZ.

Benjamin Texier (Paris 7)            Weak turbulence                              12 / 36
Weak turbulence

Issues

Propagation of chaos.
The quasi-gaussian approximation does not hold. Can it be recovered
asymptotically ?
Spatial dependence.
o
We expect the limit for the autonomous Schr¨dinger equation to be a mean-ﬁeld
model (resonant waves not localized in space, interact at all times), rather than a
collisional model such as KZ.
Existence of an invariant ensemble averaging · .
Given a P-measurable initial datum, it could be that the nonlinear dynamics takes
place in a P-null set.

Benjamin Texier (Paris 7)            Weak turbulence                              12 / 36
Weak turbulence

Issues

Propagation of chaos.
The quasi-gaussian approximation does not hold. Can it be recovered
asymptotically ?
Spatial dependence.
o
We expect the limit for the autonomous Schr¨dinger equation to be a mean-ﬁeld
model (resonant waves not localized in space, interact at all times), rather than a
collisional model such as KZ.
Existence of an invariant ensemble averaging · .
Given a P-measurable initial datum, it could be that the nonlinear dynamics takes
place in a P-null set.
More...
Stationary solutions not integrable.
Dirac mass on the resonant set not a distribution.
If discrete Fourier modes, then limiting equation is second-order in time.

Benjamin Texier (Paris 7)            Weak turbulence                              12 / 36
Weak turbulence

Context

The KZ equation is the Boltzmann equation for a microscopic model of
pseudo-particles, in a relevant mesoscopic limit.

Benjamin Texier (Paris 7)            Weak turbulence             13 / 36
Weak turbulence

Context

The KZ equation is the Boltzmann equation for a microscopic model of
pseudo-particles, in a relevant mesoscopic limit.

Description of a microscopic model in which:
Two-particle “collisions” aﬀect neither position nor momentum.
Recall cancellation property of the quadratic term above.
We can identify a cubic collision law and a Boltzmann-Grad scaling
that lead to the right limiting equation.
The Boltzmann-Grad scaling for hard spheres: Nε2 ∼ 1.
We can justify the mesoscopic limit.
Lanford, Cercignani, Illner, Pulvirenti, Gerasimenko, Petrina.

Benjamin Texier (Paris 7)            Weak turbulence               13 / 36
Weak turbulence

The microscopic model

Liouville equation:
N
∂t f +         vi ·   xi f   = 0,
i=1

with f (t, ZN ) ≥ 0,

t ≥ 0, ZN = (z1 , . . . , zN ) = (x1 , v1 , . . . , xN , vN ) ∈ DN ⊂ R6N ,

satisfying
f (t, Zσ(N) ) = f (t, ZN ),
with Zσ(N) = (xσ(1) , vσ(1) , . . . , xσ(N) , vσ(N) ), for σ ∈ SN .

Benjamin Texier (Paris 7)            Weak turbulence                              14 / 36
Weak turbulence

Distance and quasi-resonance

A collision between three particles occurs when two conditions are met:
their distance is equal to ε : d = ε
their velocities satisfy the quasi-resonance relation: r ≤ η

Benjamin Texier (Paris 7)            Weak turbulence                   15 / 36
Weak turbulence

Distance and quasi-resonance

A collision between three particles occurs when two conditions are met:
their distance is equal to ε : d = ε
their velocities satisfy the quasi-resonance relation: r ≤ η
Distance:
2                      2                    2
xj + xk                 xi + xk              xi + xj
d(xi , xj , xk ) :=         xi −                  + xj −               + xk −
2                       2                    2

Benjamin Texier (Paris 7)                Weak turbulence                                    15 / 36
Weak turbulence

Distance and quasi-resonance

A collision between three particles occurs when two conditions are met:
their distance is equal to ε : d = ε
their velocities satisfy the quasi-resonance relation: r ≤ η
Distance:
2                     2                    2
xj + xk                 xi + xk               xi + xj
d(xi , xj , xk ) :=         xi −                  + xj −                + xk −
2                       2                     2

Quasi-resonance function:
|(vi − vk ) · (vj − vk )|
r (vi , vj , vk ) := min                                ,     .
|vi − vk ||vj − vk |

Benjamin Texier (Paris 7)                  Weak turbulence                                   15 / 36
Weak turbulence

Distance and quasi-resonance

A collision between three particles occurs when two conditions are met:
their distance is equal to ε : d = ε
their velocities satisfy the quasi-resonance relation: r ≤ η
Distance:
2                     2                    2
xj + xk                 xi + xk               xi + xj
d(xi , xj , xk ) :=         xi −                  + xj −                + xk −
2                       2                     2

Quasi-resonance function:
|(vi − vk ) · (vj − vk )|
r (vi , vj , vk ) := min                                ,     .
|vi − vk ||vj − vk |

Benjamin Texier (Paris 7)                  Weak turbulence                                   15 / 36
Weak turbulence

Analogy with resonant waves

The equality r = 0 corresponds to (vi − vk ) · (vj − vk ) = 0, implying

vi + vj = vk + v
|vi |2 + |vj |2 = |vk |2 + |v |2 ,

for some v .

Resonance relation for waves with dispersion relation ω(v ) = |v |2 .

Benjamin Texier (Paris 7)                Weak turbulence               16 / 36
Weak turbulence

Domain

The subset CN of R6N in which collisions take place is

CN := ZN ∈ R6N , for some {i, j, k}, dijk = ε, rijk ≤ η ,

The domain DN is

DN := ZN ∈ R6N , for all {i, j, k},              rijk > η   or   rijk ≤ η and dijk > ε       .

Benjamin Texier (Paris 7)            Weak turbulence                            17 / 36
Weak turbulence

Pre- and post-collisional triplets

Let

d (zi , zj , zk ) := vi · (xi − xjk ) + vj · (xj − xik ) + vk · (xk − xij ).

Benjamin Texier (Paris 7)             Weak turbulence                               18 / 36
Weak turbulence

Pre- and post-collisional triplets

Let

d (zi , zj , zk ) := vi · (xi − xjk ) + vj · (xj − xik ) + vk · (xk − xij ).

˙                       ˙
Analogy with the dynamics of hard spheres: vi = xi , so that d = (2/3)d d.

Benjamin Texier (Paris 7)             Weak turbulence                               18 / 36
Weak turbulence

Pre- and post-collisional triplets

Let

d (zi , zj , zk ) := vi · (xi − xjk ) + vj · (xj − xik ) + vk · (xk − xij ).

˙                       ˙
Analogy with the dynamics of hard spheres: vi = xi , so that d = (2/3)d d.

A triplet {i, j, k} is
pre-collisional if dijk < 0
post-collisional if dijk > 0

Benjamin Texier (Paris 7)             Weak turbulence                               18 / 36
Weak turbulence

Collision law

The boundary condition for f , given on the set of conﬁgurations with a
pre-collisional triplet ({i, j, k}), is

f (. . . xi , vi , . . . xj , vj , . . . xk , vk , . . . ) = f (. . . xi , vi , . . . xj , vj , . . . xk , vk , . . . )

where

(. . . xi , vi , . . . xj , vj , . . . xk , vk , . . . ) → (. . . xi , vi , . . . xj , vj , . . . xk , vk , . . . )

is the collision law, which we require to satisfy

if dijk = ε, rijk ≤ η and dijk < 0, then
d(xi , xj , xk ) = ε, r (vi , vj , vk ) ≤ η, and d (zi , zj , zk ) > 0.

Benjamin Texier (Paris 7)                       Weak turbulence                                                  19 / 36
Weak turbulence

Collision law

Velocity collision law for exactly resonant pre-collisional triplets:

(vi , vj , vk ) → (vk , v , vm ),        m ∈ {i, j, k},

based on analogy with four-wave resonance.

Benjamin Texier (Paris 7)                  Weak turbulence                    20 / 36
Weak turbulence

Collision law

Velocity collision law for exactly resonant pre-collisional triplets:

(vi , vj , vk ) → (vk , v , vm ),        m ∈ {i, j, k},

based on analogy with four-wave resonance.
The choice
vm := vk
implying conservation of “kinetic energy”

|vi |2 + |vj |2 + |vk |2 = |vk |2 + |v |2 + |vk |2 ,

not pertinent since sign of d after collision cannot be determined from
sign of d before collision.

Benjamin Texier (Paris 7)                  Weak turbulence                    20 / 36
Weak turbulence

Collision law

Velocity collision law for exactly resonant pre-collisional triplets:

(vi , vj , vk ) → (vk , v , vm ),        m ∈ {i, j, k},

based on analogy with four-wave resonance.
The choice
vm := vk
implying conservation of “kinetic energy”

|vi |2 + |vj |2 + |vk |2 = |vk |2 + |v |2 + |vk |2 ,

not pertinent since sign of d after collision cannot be determined from
sign of d before collision.
We need a criterion to select m in {i, j}. A relevant criterion is given by the sign of
(vi − vij ) · (vk − vij ).

Benjamin Texier (Paris 7)                  Weak turbulence                            20 / 36
Weak turbulence

Collision law
Let ZN ∈ CN and {i, j, k} be a pre-collisional triplet. Up to reindexing,

|(vk − vi ) · (vk − vj )|
r (vi , vj , vk ) =                             .
|vk − vi ||vk − vj |

If (vi − vij ) · (vk − vij ) > 0, post-collisional velocities are deﬁned by
symmetry about the orthogonal plane to (vk − vi ) passing through vij :

(vi , vj , vk ) = (vk , v , vi ),

and positions are (microscopically) modiﬁed as follows, denoting by
Σkij the symmetry about the orthogonal plane to (vk − vi ) passing
through xij :
(xi , xj , xk ) = Σkij (xj , xi , xij − xk ) ;

Benjamin Texier (Paris 7)                      Weak turbulence                  21 / 36
Weak turbulence

The BBKGY hierarchy

Marginals of the distribution function:

f (s) (t, Zs ) :=             f (t, Zs , zs+1 , . . . , zN )dzs+1 · · · dzN ,
R6(N−s)

with domain

Ds := Zs = (Xs , Vs ) ∈ R6s , for all {i, j, k},

rijk > η        or   rijk ≤ η and dijk > ε     .

Benjamin Texier (Paris 7)                Weak turbulence                                 22 / 36
Weak turbulence

The BBKGY hierarchy

Integration against a test function φs :
N
∂t f +         vi ·    xi f    (t, ZN )φ(s) (t, Zs ) dZN dt = 0.
R+ ×DN                i=1

Green’s formula:

f (s) (t, Zs ) ∂t φ(s) + divXs (Vs φ(s) ) (t, Zs ) dZs dt
R+ ×Ds

=           f (s) (0, Zs )φ(s) (0, Zs ) dZs
Ds
+ Cs,s+1 (f (s+1) , φ(s) ) + Cs,s+2 (f (s+2) , φ(s) ) + Cs,s+3 (f (s+3) , φ(s) ).

Benjamin Texier (Paris 7)                    Weak turbulence                              23 / 36
Weak turbulence

Collision integrals

Cs,s+1 f (s+1) = −(N − s)                                  f (s+1) (Zs ; z)b(zi , zj , z) dz
1≤i=j≤s       R6

and
1
Cs,s+2 f (s+2) = − (N − s)(N − s − 1)
2
×                    f (s+2) (Zs ; z 1 , z 2 )b(zi , z 1 , z 2 ) dz 1 dz 2 ,
1≤i≤s   R6 ×R6

where the cross-section b is deﬁned by

d (zi , zj , zk )
b(zi , zj , zk ) = bijk :=                          1               δ                 .
d(xi , xj , xk ) r (vi ,vj ,vk )≤η d(xi ,xj ,xk )=ε

Benjamin Texier (Paris 7)                    Weak turbulence                                          24 / 36
Weak turbulence

Collision integrals

Cs,s+1 f (s+1) = −(N − s)                                  f (s+1) (Zs ; z)b(zi , zj , z) dz
1≤i=j≤s       R6

and
1
Cs,s+2 f (s+2) = − (N − s)(N − s − 1)
2
×                    f (s+2) (Zs ; z 1 , z 2 )b(zi , z 1 , z 2 ) dz 1 dz 2 ,
1≤i≤s   R6 ×R6

where the cross-section b is deﬁned by

d (zi , zj , zk )
b(zi , zj , zk ) = bijk :=                          1               δ                 .
d(xi , xj , xk ) r (vi ,vj ,vk )≤η d(xi ,xj ,xk )=ε
˜
The collision integral Cs,s+3 vanishes by symmetry.
Benjamin Texier (Paris 7)                    Weak turbulence                                          24 / 36
Weak turbulence

Gain and loss collision integrals
The collision cross-section splits into

b(zi , zj , zk ) = bijk 1dijk >0 − bijk 1dijk <0 =: b+ − b− .
ijk  ijk

corresponding to post- and pre-collisional conﬁgurations.

Benjamin Texier (Paris 7)                   Weak turbulence                          25 / 36
Weak turbulence

Gain and loss collision integrals
The collision cross-section splits into

b(zi , zj , zk ) = bijk 1dijk >0 − bijk 1dijk <0 =: b+ − b− .
ijk  ijk

corresponding to post- and pre-collisional conﬁgurations.

Cs,s+1 f (s+1) = (N − s)

×                        f (s+1) (Zs ; z ) − f (s+1) (Zs ; z) b+ (zi , zj , z ) dz
1≤i=j≤s         R6

1
Cs,s+2 f (s+2) = (N − s)(N − s − 1)
2
×                          f (s+2) (Zs ; (z 1 ) , (z 2 ) ) − f (s+2) (Zs ; z 1 , z 2 )
1≤i≤s      R6 ×R6

× b+ (zi , z 1 , z 2 ) dz 1 dz 2 ,
where Z → Z is the collision law.
Benjamin Texier (Paris 7)                    Weak turbulence                                    25 / 36
Weak turbulence

Integral formulation of the BBKGY hierarchy

t
f (s) (t) = f (s) (0) +             Ss (t − τ ) Cs,s+1 f (s+1) + Cs,s+2 f (s+2) (τ ) dτ
0

where S is the transport operator

S(t) :              f → f (Z(−t, ·)),

with Z particle transport in DN , and Ss restriction to R × Ds .

Benjamin Texier (Paris 7)                    Weak turbulence                              26 / 36
Weak turbulence

Iterated time integration

∞          t       t1             tn−1
f (s) (t) =                             ...
n=0     0       0              0          ki+1 −ki =1,2

Ss (t − t1 )Cs,s+k1 Ss+k1 (t1 − t2 )Cs+k1 ,s+k2
. . . Cs+kn−1 ,s+kn Ss+kn (tn )f (s+kn ) (0)dtn . . . dt1
(2)
where by convention f (s) (0) ≡ 0 for s > N.

Benjamin Texier (Paris 7)                           Weak turbulence               27 / 36
Weak turbulence

The associated Boltzmann hierarchy

∞          t        t1              tn−1
(s)                                                        0           0      0              0
f0 (t) =                                 ...               Ss (t − t1 )Cs,s+2 Ss+2 (t1 − t2 )Cs+2,s+4 . . .
n=0     0        0               0
0          (s+2n)
(0)dtn . . . dt1
. . . Ss+2n (tn )f0
(3)
(s)                     0 free transport operators and C 0
where f0 (0) s∈N initial datum, Ss                                s,s+2
the scaled collision operators
1
Cs,s+2 f (s+2) =
0
f (s+2) (Zs∗ ; (z 1 )∗ , (z 2 )∗ ) − f (s+2) (Zs ; z 1 , z 2 )
2                   R12
1≤i≤s
+
× bR (zi , z 1 , z 2 )dz 1 dz 2 .

Benjamin Texier (Paris 7)                                Weak turbulence                                       28 / 36
Weak turbulence

Assumption on the datum

For a family f (0) = f s (0)            s∈N∗
of functions in L∞ (R6s ; R+ ), we consider
the following assumptions:

for some R ≥ 1,                for all s,       f (s) (0) is supported in |Vs | ≤ R,           (4)

where |Vs | := sup |vi |, and
1≤i≤s

∀n ∈ N∗ ,         f (s) (0, Zs ) =        f (s+n) (0, Zs , zs+1 , . . . , zs+n )dzs+1 . . . dzs+n ,
(5)
with the uniform bound

f (0)   L∞   := sup f (s) (0)          L∞ (R6s )   < ∞.                 (6)
s∈N∗

Benjamin Texier (Paris 7)                    Weak turbulence                                   29 / 36
Weak turbulence

We consider N → ∞, ε → 0 and η → 0 with

α := N 2 ε5 η ∼ 1,          β := Nε2 η = O(1).   (7)

Benjamin Texier (Paris 7)               Weak turbulence               30 / 36
Weak turbulence

Result: existence for the BBKGY and Boltzmann hierarchy

Theorem (Existence)

Consider a family f (0) = f s (0) s∈N∗ of initial data in L∞ (R6s ; R+ )
satisfying (4), (5) and (6). For some t ∗ > 0, in the scaling (7), the
BBGKY hierarchy (2) and the Boltzmann hierarchy (3) with respective
initial datum (f (s) (0))1≤s≤N and f (0) have unique mild
(s)
solutions (f (s) )1≤s≤N and (f0 )s∈N∗ over [0, t∗ ), which satisfy the bounds
n
f (s) (t)   L∞   ≤ f (0)      L∞            CR 6 max(α, β)t ,
n≥0
(s)                                              n
f0 (t) L∞        ≤ f (0)      L∞            CR 6 αt .
n≥0

for some C > 0 which depends neither on f (0), nor on N, ε, η.

Benjamin Texier (Paris 7)                   Weak turbulence                       31 / 36
Weak turbulence

Result: term-by-term convergence

Theorem (Convergence)

Consider a family f (0) = f s (0) s∈N∗ of initial data in the
space L∞ (R6s ; R+ ) satisfying (4), (5) and (6). For the same t∗ > 0 as in
Theorem 1.1, in the scaling (7), the mild solution to the BBGKY hierarchy
(2) with initial datum (f (s) (0))1≤s≤N converges to the mild solution of the
Boltzmann hierarchy (3) with initial datum f (0) on [0, t ∗ ), in the sense
(s)
that for all s, f (s) → f0 on [0, t ∗ ), uniformly in Xs on the complement of
a set of arbitrarily small measure, and tightly in Vs .

Benjamin Texier (Paris 7)            Weak turbulence                  32 / 36
Weak turbulence

Propagation of chaos in the mesoscopic limit

Solutions of the Boltzmann hierarchy issued from factorized initial data
(s)     ⊗s
f0 (0) = F0 ,                          (8)

are factorized:
(s)
f0 (t) = F ⊗s (t),                       (9)
and the ﬁrst marginal F = F (t, x, v ) solves the KZ equation (with a
¯
cross-section bR which depends on our microscopic model).

Benjamin Texier (Paris 7)              Weak turbulence                  33 / 36
Weak turbulence

Convergence proof

(s)
Goal is to show f (s) → f0 .

Benjamin Texier (Paris 7)                  Weak turbulence   34 / 36
Weak turbulence

Convergence proof

(s)
Goal is to show f (s) → f0 .
Decomposition into elementary collision terms:

Es,n (t, K , J, M, T )
j1 ,m1                  j2 ,m              j   ,m
n−1
:= |J|Ss (t − t1 )Cs,s+k1 Ss+k1 (t1 − t2 )Cs+k12,s+k2 . . . Cs+kn−1n−1 n Ss+kn (tn )
,s+k

Benjamin Texier (Paris 7)                  Weak turbulence                      34 / 36
Weak turbulence

Convergence proof

(s)
Goal is to show f (s) → f0 .
Decomposition into elementary collision terms:

Es,n (t, K , J, M, T )
j1 ,m1                  j2 ,m                     j   ,m
n−1
:= |J|Ss (t − t1 )Cs,s+k1 Ss+k1 (t1 − t2 )Cs+k12,s+k2 . . . Cs+kn−1n−1 n Ss+kn (tn )
,s+k

Given a conﬁguration Xs ∈ Ds , decompose

Es,n (t, K , J, M, T )f (s+kn ) (0, Xs , Vs )φs (Vs )

into “good” terms (including only Cs,s+2 collision operators) and “bad
terms” associated with recollisions.

Benjamin Texier (Paris 7)                  Weak turbulence                      34 / 36
Weak turbulence

Pathological trajectories
Size (in Vs space) of “pathological” trajectories associated with
recollisions based on geometrical lemma:
Lemma
Let 0 < ε < ε0 . Given two positions x1 and x2 such that

|x1 − x2 | ≥ ε0 ,

and one velocity v1 ∈ B(0, R), there exists a set M ⊂ B(0, R), the
measure of which is controlled as follows
2
ε
|M| ≤ C
ε0

and such that for any v2 ∈ B(0, R) \ M

∀τ > 0,    |(x1 − v1 τ ) − (x2 − v2 τ )| > ε .
Benjamin Texier (Paris 7)                Weak turbulence               35 / 36
Weak turbulence

Time of validity

Mean free path for hard spheres = 1/(Nε2 ).

Benjamin Texier (Paris 7)            Weak turbulence   36 / 36
Weak turbulence

Time of validity

Mean free path for hard spheres = 1/(Nε2 ).

The time of validity is a fraction of α :

Benjamin Texier (Paris 7)            Weak turbulence   36 / 36
Weak turbulence

Time of validity

Mean free path for hard spheres = 1/(Nε2 ).

The time of validity is a fraction of α :

t∗ = a fraction of the mean time between two collisions.

Benjamin Texier (Paris 7)            Weak turbulence                36 / 36
Weak turbulence

Time of validity

Mean free path for hard spheres = 1/(Nε2 ).

The time of validity is a fraction of α :

t∗ = a fraction of the mean time between two collisions.

There exists a threshold αc < 1, such that as N goes to inﬁnity and under
the assumption that there are at most αN interactions with α < αc , then
collisions take place almost surely between uncorrelated particles.

Benjamin Texier (Paris 7)            Weak turbulence                36 / 36
Weak turbulence

Time of validity

Mean free path for hard spheres = 1/(Nε2 ).

The time of validity is a fraction of α :

t∗ = a fraction of the mean time between two collisions.

There exists a threshold αc < 1, such that as N goes to inﬁnity and under
the assumption that there are at most αN interactions with α < αc , then
collisions take place almost surely between uncorrelated particles.

This decorrelation property breaks down if α ≥ αc .

Benjamin Texier (Paris 7)            Weak turbulence                36 / 36

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