APPENDIX C SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL

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APPENDIX C : SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL In this appendix, we use the notations of chapter 2. We shall give several formulas which hold for Landau's collisional operator Q(f, f) with a Coulomb potential. We assume that f is smooth and rapidly decreasing, so as to allow all formal manipulations. In this case, the matrix aij is given by (1) aij(v) = ( ?ij ? vivj |v|2 ) 1 |v| . Therefore, c = ∑ ij ∂ijaij = ?8pi?, where ? is the Dirac mass at the origin, and hence c = c ? f = ?8pif . Thus, the Landau operator can be rewritten as (2) Q(f, f) = aij∂ijf + 8pif 2. As pointed out to us to by Desjardins, the matrix aij can be com- puted “eplicitly” by the use of the so-called Riesz transform. Let us define f?(?) = ∫ R3 e?iv·?f(v) dv, Rij = ?∂ij(?∆)?1, so that in Fourier space, Rij is given by R?ij(?) = ?i?j |?|2 , and Rij satisfies the identities ∑ i Rii = I, ∂ij = Rij∆ = ∆Rij, ∑ ij Rij∂ij = ∆.

?2 ∫

another formal

dirac function

fourier transform

r6 aij

lan- dau operator

r3 cf

ij ∂ijaij

physically rel- evant

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Publié par

profil-urra-2012

Nombre de lectures

Langue

English

Dirac delta function Fourier transform

APPENDIX C : SOME IDENTITIES SATISFIED BY THE COULOMB POTENTIAL

In this appendix, we use the notations of chapter 2.We shall give several formulas which hold for Landau’s collisional operatorQ(f, f) with a Coulomb potential.We assume thatfis smooth and rapidly decreasing, so as to allow all formal manipulations. In this case, the matrixaijis given by   vivj1 (1)aij(v) =ij. 2 |v| |v| P Therefore,c=∂ijaij=8 , whereis the Dirac mass at the ij origin, and hencec=cf=8 f. Thus,the Landau operator can be rewritten as 2 (2)Q(f, f) =aij∂ijf+ 8. f As pointed out to us to by Desjardins, the matrixaijcan be com-puted “eplicitly” by the use of the so-called Riesz transform.Let us dene Z iv b f() =e f(v)dv, 3 R 1 Rij=∂ij(), so that in Fourier space,Rijis given by ij b Rij() =, 2 || andRijdiehtsesitasestitien X X Rii=I, ∂ij=Rij =Rij, Rij∂ij= . i ij It turns out that ij abij() = 8 , 4 || so that 1 (3)aij=aijf=8 Rijf. Hence the equation (2) can be rewritten as (4)    1 21 2 Q(f, f) = 8(Rijf)∂ijf+f= 8(Rijf)(Rijf) +f . 1

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