logo AFST
Discrete coagulation-fragmentation system with transport and diffusion
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 439-460.

On démontre l’existence de solutions pour deux systèmes infinis d’équations de coagulation-fragmentation. Dans un premier cas, on rajoute un terme de transport au système classique de coagulation-fragmentation et dans un second cas on rajoute un terme de transport et un terme de diffusion. Dans les deux cas les particules possèdent la même vitesse que le fluide et dans le second cas les coefficients de diffusion sont égaux. On résout dans un premier temps un problème tronqué en taille puis on passe à la limite en utilisant des lemmes de compacité

We prove the existence of solutions to two infinite systems of equations obtained by adding a transport term to the classical discrete coagulation-fragmentation system and in a second case by adding transport and spacial diffusion. In both case, the particles have the same velocity as the fluid and in the second case the diffusion coefficients are equal. First a truncated system in size is solved and after we pass to the limit by using compactness properties.

Reçu le : 2007-03-01
Accepté le : 2008-04-09
Publié le : 2010-12-06
DOI : https://doi.org/10.5802/afst.1189
@article{AFST_2008_6_17_3_439_0,
     author = {St\'ephane Brull},
     title = {Discrete coagulation-fragmentation system with transport and diffusion},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
     number = {3},
     year = {2008},
     pages = {439-460},
     doi = {10.5802/afst.1189},
     zbl = {1158.76041},
     mrnumber = {2488228},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2008_6_17_3_439_0/}
}
Stéphane Brull. Discrete coagulation-fragmentation system with transport and diffusion. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 439-460. doi : 10.5802/afst.1189. https://afst.centre-mersenne.org/item/AFST_2008_6_17_3_439_0/

[1] Aman (H.).— Coagulation-fragmentation processes Arch. Ration. Mech. Anal. 151, p. 339-366 (2000). | MR 1756908 | Zbl 0977.35060

[2] Ball (J.M.), Carr (J.).— The discrete coagulation-fragmentation equations : existence, uniqueness and density conservation. Journ.stat.phys., 61, p. 203-234 (1990). | MR 1084278

[3] Ball (J.M.), Carr (J.), Penrose (O.).— The Becker-Doring Cluster Equations : Basic Properties and asymptotic Behaviour of Solutions. Comm. Math. Phys., 104, p. 657-692 (1986). | MR 841675 | Zbl 0594.58063

[4] Burobin (A.V.).— Existence and uniqueness of a solution of a Cauchy Problem for inhomogeneous three-dimentional coagulation. differential equations 19, p. 1187-1197 (1983). | MR 718559 | Zbl 0553.35072

[5] Chae (D.), Dubovskii (P.).— Existence and uniqueness for spacially inhomogeneous coagulation-condensation equation with unbounded kernels Journ. of integral equations vol 9, No 3, p. 279-236 (1997). | MR 1616462 | Zbl 0907.45010

[6] Collet (J.F.), Poupaud (F.).— Asymptotic behaviour of solutions to the diffusive fragmentation-coagulation system. Physica D vol.114, p. 123-146 (1998). | MR 1612051 | Zbl 0960.82017

[7] Collet (J.F.), Poupaud (F.).— Existence of solutions to coagulation-fragmentation systems with diffusion. Transp. Theory. Stat. Phys. 25, p. 503-513 (1996). | MR 1407550 | Zbl 0870.35117

[8] Dubovskii (P.B.).— Existence theorem for space inhomogeneous coagulation equation. Differential equations 26, p. 508-513 (1990).

[9] Guyon (E.).— Hydrodynamique Physique. édition EDP Sciences (2001).

[10] Laurencot (P.), Mischler (S.).— Global existence for the discrete diffusive coagulation-fragmentation equations in L 1 . Rev.Mat.Iberoaericana 18, p. 221-235 (2002). | MR 1954870 | Zbl 1036.35089

[11] Laurencot (P.), Mischler (S.).— The continous coagulation-fragmentation equations with diffusion. Arch.Rational Mech. Anal. 162 No1, p. 45-99 (2002). | MR 1892231 | Zbl 0997.45005

[12] Laurencot (P.), Mischler (S.).— From the discrete to the continous coagulation-fragmentation equations. Proc. Roy. Soc. Edinburgh 132A, p. 1219-1248 (2002). | MR 1938720 | Zbl 1034.35011

[13] Simon (J.).— Compact sets in L p (0,T,B), Ann. Mat. Pura. Appl., IV 146, p. 65-96 (1987). | MR 916688 | Zbl 0629.46031

[14] Slemrod (M.).— Coagulation-diffusion : derivation and existence of solutions for the diffuse interface stucture equations. Physica D, 46 (3), p. 351-366 (1990). | MR 1081687 | Zbl 0732.35103