Equilibria for collisions kernels appearing in weak turbulence - - PowerPoint PPT Presentation

Equilibria for collisions kernels appearing in weak turbulence

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique, October 22, 2019

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Boltzmann operator of rarefied gases

Q(f )(v) =

• R3
• S2
• f (v + σ · (v∗ − v) σ) f (v∗ − σ · (v∗ − v) σ)

−f (v) f (v∗)

• B dσdv∗

Abstract formulation: Q(f )(v) =

• R3
• R3
• R3
• f (v ′) f (v ′

∗)

−f (v) f (v∗)

• B

× δ{v+v∗=v′+v′

∗} δ{|v|2+|v∗|2=|v′|2+|v′ ∗|2} dv∗dv ′

∗dv ′.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

First part of the H-theorem of Boltzmann

Entropy: H(f ) := (−)

• R3 f (v) ln f (v) dv;

Entropy production:

• R3 Q(f )(v) ln f (v) dv = 1

4

• R3
• R3
• R3
• R3
• ln f (v) + ln f (v∗) − ln f (v ′) − ln f (v ′

∗)

• × [f (v)f (v∗)−f (v ′)f (v ′

∗)]δ{v+v∗=v′+v′

∗} δ{|v|2+|v∗|2=|v′|2+|v′ ∗|2} dvdv∗dv ′

∗dv ′.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Definition of equilibria

Definition: the equilibria of the Boltzmann equation are the functions f > 0 such that when v + v∗ = v ′ + v ′

∗

and |v|2 + |v∗|2 = |v ′|2 + |v ′

∗|2,

• ne has

f (v)f (v∗) − f (v ′)f (v ′

∗),

• r equivalently, for g = ln f ,

g(v ′) + g(v ′

∗) = g(v) + g(v∗).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: explicit form of equilibria

It is clear that for all a, c ∈ R, b ∈ R3, the function g(v) := a + b · v + c|v|2 is an equilibrium. Second part of H-theorem of Boltzmann: All equilibria (in a suitable functional space) have this form. Natural functional space for g: weighted L2.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: traditional proof (i)

For an equilibrium g, g(v + σ · (v∗ − v) σ) + g(v∗ − σ · (v∗ − v) σ) = g(v) + g(v∗), which, selecting σ · (v∗ − v) ∼ 0, leads to (∇g(v) − ∇g(v∗)) × (v − v∗) = 0,

• r in coordinates (for i = j)

(∂jg(v) − ∂jg(v∗)) (vi − vi∗) = (∂ig(v) − ∂ig(v∗)) (vj − vj∗).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: traditional proof (ii)

The formula (∇g(v) − ∇g(v∗)) × (v − v∗) = 0 corresponds to equilibria of the Landau equation for collisions in ionized plasmas (Coulomb interaction) QL(f )(v) = ∇ ·

• R3
• Id − v − v∗

|v − v∗| ⊗ v − v∗ |v − v∗|

• ∇ ln f (v) − ∇ ln f (v∗)
• f (v) f (v∗) |v − v∗|−1 dvdv∗.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: traditional proof (iii)

New starting point (for i = j) (∂jg(v) − ∂jg(v∗)) (vi − vi∗) = (∂ig(v) − ∂ig(v∗)) (vj − vj∗). After applying ∂vi∗, −∂ijg(v∗) (vi − vi∗) − (∂jg(v) − ∂jg(v∗)) = −∂iig(v∗) (vj − vj∗). After applying ∂vi on one hand, and ∂vj on the other hand, −∂ijg(v∗) − ∂ijg(v) = 0, −∂jjg(v) = −∂iig(v∗).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: traditional proof (iv)

New starting point (for i = j) −∂ijg(v∗) − ∂ijg(v) = 0, −∂jjg(v) = −∂iig(v∗). So Hess(g) = 2c Id, and g(v) := a + b · v + c|v|2. Can be applied in the classical sense if g is of class C 2, and in the sense

• f distributions if g lies in L1

loc (that is, using integrations in v, v∗).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Boltzmann operator for the four waves equation of weak turbulence theory (Zakharov)

QW (f )(v) =

• Rd
• Rd
• Rd W (v, v∗, v ′, v ′

∗)

• f (v ′) f (v ′

∗) (f (v) + f (v∗))

−f (v) f (v∗) (f (v ′) + f (v ′

∗))

• × δ{v+v∗=v′+v′

∗} δ{ω(v)+ω(v∗)=ω(v′)+ω(v′ ∗)} dv∗dv ′

∗dv ′.

Typical ω: ω(v) = C |v|α, for 0 < α < 1 and C > 0. In particular, in the two-dimensional case, ω(v) = C

• |v| is used to

describe gravitational waves on a fluid surface

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

First part of the H-theorem for the 4-waves operator

Entropy: H(f ) :=

• Rd ln f (v) dv;

Entropy production:

• Rd QW (f )(v) f −1(v) dv = 1

4

• W (v, v∗, v ′, v ′

∗)

×

• f −1(v) + f −1(v∗) − f −1(v ′) − f −1(v ′

∗)

2 × f (v)f (v∗)f (v ′)f (v ′

∗)δ{v+v∗=v′+v′

∗} δ{ω(v)+ω(v∗)=ω(v′)+ω(v′ ∗)} dvdv∗dv ′

∗dv ′.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Definition of equilibria

Definition: the equilibria of the 4-waves equation are the functions f > 0 such that when v + v∗ = v ′ + v ′

∗

and ω(v) + ω(v∗) = ω(v ′) + ω(v ′

∗),

• ne has

f −1(v ′) + f −1(v ′

∗) = f −1(v) + f −1(v∗),

• r equivalently, for g = f −1,

g(v ′) + g(v ′

∗) = g(v) + g(v∗).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Second part of H-theorem: explicit form of equilibria

It is clear that for all a, c ∈ R, b ∈ Rd, the function g(v) := a + b · v + cω(v) is an equilibrium. Expected result (Second part of H-theorem): All equilibria (in a suitable functional space) have this form [except maybe for a small class of functions ω]. Natural functional space for g: weighted L2.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Existing results

Case ω(v) = |v|2 (Boltzmann equation for monoatomic gases) : Proof when g is C 2 (Boltzmann); Proof when g is measurable, or a distribution (Truesdell-Muncaster; Wennberg) Case ω(v) =

• 1 + |v|2 (Boltzmann equation for relativistic monoatomic

gases) : Proof when g is C 2 (Cercignani, Kremer); Proof when g is a distribution (suggested in Cercignani, Kremer)

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Result in the general case

Theorem (Breden, LD): Let d ∈ {2, 3} and ω ∈ C 2(Rd − {0}). Assume that there exist i, j ∈ {1, . . . , d}, i = j, such that {1, ∂iω, ∂jω} are linearly independant in C 1(Rd − {0}). Assume also that the boundary ∂A of A :=

• (v, v∗) ∈
• Rd2 , ∇ω(v) = ∇ω(v∗)
• .

is of measure 0 in

• Rd2.

Let g ∈ L1

loc(Rd) be an equilibrium.

Then, there exist a, c ∈ R and b ∈ Rd such that, for a.e. v in Rd, g(v) = a + b · v + c ω(v).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Method of proof: first step

As in the proof in the Boltzmann case, we consider only grazing collisions, that is, collisions for which v ′ ∼ v, v ′

∗ ∼ v∗.

Then, the equilibria satisfy the following property (for a.e. v, v∗ ∈ Rd): (∇g(v) − ∇g(v∗)) × (∇ω(v) − ∇ω(v∗)) = 0,

• r in coordinates (for i = j)

(∂jg(v)−∂jg(v∗)) (∂iω(v)−∂iω(v∗)) = (∂ig(v)−∂ig(v∗)) (∂jω(v)−∂jω(v∗)). This amounts to say that the entropy dissipation of the grazing collision approximation (Landau-type operator) of QW is zero.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Method of proof: second step

Multiplying (the i, j-th component of) the identity (∂jg(v)−∂jg(v∗)) (∂iω(v)−∂iω(v∗)) = (∂ig(v)−∂ig(v∗)) (∂jω(v)−∂jω(v∗)). by β(v∗) and integrating w.r.t. v∗, we get

Rd β(v∗)dv∗

− [∂ig](v)[∂jω](v) + [∂jg](v)[∂iω](v)

• −

Rd β(v∗)∂iω(v∗)dv∗

• ∂jg(v) +

Rd β(v∗)∂jω(v∗)dv∗

• ∂ig(v)

=

Rd g(v∗)∂iβ(v∗)dv∗

• ∂jω(v)dv −

Rd g(v∗)∂jβ(v∗)dv∗

• ∂iω(v)dv

−

Rd g(v∗) (∂iβ(v∗)∂jω(v∗) − ∂jβ(v∗)∂iω(v∗)) dv∗

• .

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Method of proof: second step (II)

Multiplying then (the i, j-th component of) the same identity by β(v∗) ∂iω(v∗) and integrating w.r.t. v∗, we get

Rd β(v∗)∂iω(v∗)dv∗

− [∂ig](v)[∂jω](v) + [∂jg](v)[∂iω](v)

• −

Rd β(v∗) (∂iω(v∗))2 dv∗

• ∂jg(v)+

Rd β(v∗)∂iω(v∗)∂jω(v∗)dv∗

• ∂ig(v)

=

Rd g(v∗)

• ∂2

iiω(v∗)β(v∗) + ∂iω(v∗)∂iβ(v∗)

• dv∗
• ∂jω(v)dv

−

Rd g(v∗)

• ∂2

ijω(v∗)β(v∗) + ∂iω(v∗)∂jβ(v∗)

• dv∗
• ∂iω(v)dv

−

• Rd g(v∗)
• (∂2

iiω(v∗)β(v∗) + ∂iω(v∗)∂iβ(v∗))∂jω(v∗)

−(∂2

ijω(v∗)β(v∗) + ∂iω(v∗)∂jβ(v∗))∂iω(v∗)

• dv∗.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Method of proof: second step (III)

Cramer 3 × 3 system M u = w, with M =  

• Rd β
• Rd ∂iωβ
• Rd ∂jω(v∗)β
• Rd ∂iωβ
• Rd(∂iω)2β
• Rd ∂iω∂jωβ
• Rd ∂jωβ
• Rd ∂iω∂jωβ
• Rd(∂jω)2β

  , u =   −[∂ig](v)[∂jω](v) + [∂jg](v)[∂iω](v) −∂jg(v) ∂ig(v)   , and w =   (k1 + k2∂iω(v) + k3∂jω(v)) (k4 + k5∂iω(v) + k6∂jω(v)) (k7 + k8∂iω(v) + k9∂jω(v))   .

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Method of proof: second step (IV)

We see that ∂ig(v) = [DetM]−1 (k7 + k8∂iω(v) + k9∂jω(v)) . Then, after some algebra, k9 = 0 (and −k6 = k8), so that by taking primitives, g(v) = [DetM]−1 (k0 + k7 · v + k8ω(v)). The system satisfies (for some well chosen function β, and i, j ∈ {1, . . . , d} ) DetM = 0 because of the assumption: There exist i, j ∈ {1, . . . , d}, i = j, such that {1, ∂iω, ∂jω} are linearly independant in C 1(Rd − {0}).

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Remarks

The assumption: There exist i, j ∈ {1, . . . , d}, i = j, such that {1, ∂iω, ∂jω} are linearly independant in C 1(Rd − {0}). is close to optimal: when it is not satified for d = 2, counter-examples exist. The method of proof is based on ideas taken from the study of Cercignani’s conjecture for Landau’s equation with Coulomb potential, cf. LD 2015; LD; Carrapatoso, LD, He 2017, Suboptimal result of the same type for the 3-waves equation.

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,

Perspectives

Better result for the 3-waves equation Spectral gaps for the 3-waves and 4-waves linearized equations

Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,