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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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