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 ) = f ( v + σ · ( v ∗ − v ) σ ) f ( v ∗ − σ · ( v ∗ − v ) σ ) R 3 S 2 � − f ( v ) f ( v ∗ ) B d σ dv ∗ Abstract formulation: � � � � f ( v ′ ) f ( v ′ Q ( f )( v ) = ∗ ) R 3 R 3 R 3 � − f ( v ) f ( v ∗ ) B ∗ | 2 } dv ∗ dv ′ ∗ dv ′ . × δ { v + v ∗ = v ′ + v ′ ∗ } δ {| v | 2 + | v ∗ | 2 = | v ′ | 2 + | v ′ 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 ) := ( − ) R 3 f ( v ) ln f ( v ) dv ; Entropy production : R 3 Q ( f )( v ) ln f ( v ) dv = 1 � 4 � � � � � � ln f ( v ) + ln f ( v ∗ ) − ln f ( v ′ ) − ln f ( v ′ ∗ ) R 3 R 3 R 3 R 3 × [ f ( v ) f ( v ∗ ) − f ( v ′ ) f ( v ′ ∗ | 2 } dvdv ∗ dv ′ ∗ dv ′ . ∗ )] δ { v + v ∗ = v ′ + v ′ ∗ } δ {| v | 2 + | v ∗ | 2 = | v ′ | 2 + | v ′ 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 , one has f ( v ) f ( v ∗ ) − f ( v ′ ) f ( v ′ ∗ ) , or 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 ∈ R 3 , 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 L 2 . 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 , or in coordinates (for i � = j ) ( ∂ j g ( v ) − ∂ j g ( v ∗ )) ( v i − v i ∗ ) = ( ∂ i g ( v ) − ∂ i g ( v ∗ )) ( v j − v j ∗ ) . 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) � � Id − v − v ∗ | v − v ∗ | ⊗ v − v ∗ �� � Q L ( f )( v ) = ∇ · ∇ ln f ( v ) − ∇ ln f ( v ∗ ) | v − v ∗ | R 3 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 ) ( ∂ j g ( v ) − ∂ j g ( v ∗ )) ( v i − v i ∗ ) = ( ∂ i g ( v ) − ∂ i g ( v ∗ )) ( v j − v j ∗ ) . After applying ∂ v i ∗ , − ∂ ij g ( v ∗ ) ( v i − v i ∗ ) − ( ∂ j g ( v ) − ∂ j g ( v ∗ )) = − ∂ ii g ( v ∗ ) ( v j − v j ∗ ) . After applying ∂ v i on one hand, and ∂ v j on the other hand, − ∂ ij g ( v ∗ ) − ∂ ij g ( v ) = 0 , − ∂ jj g ( v ) = − ∂ ii g ( 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 ) − ∂ ij g ( v ∗ ) − ∂ ij g ( v ) = 0 , − ∂ jj g ( v ) = − ∂ ii g ( v ∗ ) . So Hess ( g ) = 2 c 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 of distributions if g lies in L 1 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) � � � � R d W ( v , v ∗ , v ′ , v ′ f ( v ′ ) f ( v ′ Q W ( f )( v ) = ∗ ) ∗ ) ( f ( v ) + f ( v ∗ )) R d R d � − f ( v ) f ( v ∗ ) ( f ( v ′ ) + f ( v ′ ∗ )) ∗ ) } dv ∗ dv ′ ∗ dv ′ . × δ { v + v ∗ = v ′ + v ′ ∗ } δ { ω ( v )+ ω ( v ∗ )= ω ( v ′ )+ ω ( v ′ 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 ) := R d ln f ( v ) dv ; Entropy production : � R d Q W ( f )( v ) f − 1 ( v ) dv = 1 � W ( v , v ∗ , v ′ , v ′ ∗ ) 4 � 2 � f − 1 ( v ) + f − 1 ( v ∗ ) − f − 1 ( v ′ ) − f − 1 ( v ′ × ∗ ) × f ( v ) f ( v ∗ ) f ( v ′ ) f ( v ′ ∗ ) } dvdv ∗ dv ′ ∗ dv ′ . ∗ ) δ { v + v ∗ = v ′ + v ′ ∗ } δ { ω ( v )+ ω ( v ∗ )= ω ( v ′ )+ ω ( v ′ 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 ′ ∗ ) , one has f − 1 ( v ′ ) + f − 1 ( v ′ ∗ ) = f − 1 ( v ) + f − 1 ( v ∗ ) , or 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 ∈ R d , the function g ( v ) := a + b · v + c ω ( 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 L 2 . 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) 1 + | v | 2 (Boltzmann equation for relativistic monoatomic � Case ω ( v ) = 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 ( R d − { 0 } ). Assume that there exist i , j ∈ { 1 , . . . , d } , i � = j , such that { 1 , ∂ i ω, ∂ j ω } are linearly independant in C 1 ( R d − { 0 } ) . Assume also that the boundary ∂ A of R d � 2 , ∇ ω ( v ) � = ∇ ω ( v ∗ ) � � � A := ( v , v ∗ ) ∈ . R d � 2 . � is of measure 0 in Let g ∈ L 1 loc ( R d ) be an equilibrium. Then, there exist a , c ∈ R and b ∈ R d such that, for a.e. v in R d , 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 ∗ ∈ R d ): ( ∇ g ( v ) − ∇ g ( v ∗ )) × ( ∇ ω ( v ) − ∇ ω ( v ∗ )) = 0 , or in coordinates (for i � = j ) ( ∂ j g ( v ) − ∂ j g ( v ∗ )) ( ∂ i ω ( v ) − ∂ i ω ( v ∗ )) = ( ∂ i g ( v ) − ∂ i g ( v ∗ )) ( ∂ j ω ( v ) − ∂ j ω ( v ∗ )) . This amounts to say that the entropy dissipation of the grazing collision approximation (Landau-type operator) of Q W is zero. Laurent Desvillettes, Univ. Paris Diderot, IMJ-PRG, in collaboration with Maxime Breden, Ecole Polytechnique,
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