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.
@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},
pages = {439--460},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 17},
number = {3},
year = {2008},
doi = {10.5802/afst.1189},
mrnumber = {2488228},
zbl = {1158.76041},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1189/}
}
TY - JOUR AU - Stéphane Brull TI - Discrete coagulation-fragmentation system with transport and diffusion JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2008 SP - 439 EP - 460 VL - 17 IS - 3 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1189/ DO - 10.5802/afst.1189 LA - en ID - AFST_2008_6_17_3_439_0 ER -
%0 Journal Article %A Stéphane Brull %T Discrete coagulation-fragmentation system with transport and diffusion %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2008 %P 439-460 %V 17 %N 3 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1189/ %R 10.5802/afst.1189 %G en %F 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/articles/10.5802/afst.1189/
[1] Aman (H.).— Coagulation-fragmentation processes Arch. Ration. Mech. Anal. 151, p. 339-366 (2000). | MR | Zbl
[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
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 . Rev.Mat.Iberoaericana 18, p. 221-235 (2002). | MR | Zbl
[11] Laurencot (P.), Mischler (S.).— The continous coagulation-fragmentation equations with diffusion. Arch.Rational Mech. Anal. 162 No1, p. 45-99 (2002). | MR | Zbl
[12] Laurencot (P.), Mischler (S.).— From the discrete to the continous coagulation-fragmentation equations. Proc. Roy. Soc. Edinburgh 132A, p. 1219-1248 (2002). | MR | Zbl
[13] Simon (J.).— Compact sets in , Ann. Mat. Pura. Appl., IV 146, p. 65-96 (1987). | MR | Zbl
[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 | Zbl
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