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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);
    it admits heavy-tailed stationary solutions

                               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 |

The link between N, ε and η is the Boltzmann-Grad scaling.

   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


The Boltzmann-Grad scaling




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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