Uniqueness of post-gelation solutions of a class of coagulation equations
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, p. 189-215
We prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition. To this end, we solve an associated partial differential equation involving the generating functions before and after the phase transition. Applications include the classical Smoluchowski and Flory equations with multiplicative coagulation rate and the recently introduced symmetric model with limited aggregations. For the latter, we compute the limiting concentrations and we relate them to random graph models.
DOI : https://doi.org/10.1016/j.anihpc.2010.10.005
Classification:  34A34,  82D60
Keywords: Coagulation equations, Gelation, Generating functions, Method of characteristics, Long-time behavior
@article{AIHPC_2011__28_2_189_0,
     author = {Normand, Raoul and Zambotti, Lorenzo},
     title = {Uniqueness of post-gelation solutions of a class of coagulation equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {2},
     year = {2011},
     pages = {189-215},
     doi = {10.1016/j.anihpc.2010.10.005},
     zbl = {1213.82116},
     mrnumber = {2784069},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_2_189_0}
}
Normand, Raoul; Zambotti, Lorenzo. Uniqueness of post-gelation solutions of a class of coagulation equations. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 2, pp. 189-215. doi : 10.1016/j.anihpc.2010.10.005. http://www.numdam.org/item/AIHPC_2011__28_2_189_0/

[1] D.J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli 5 no. 1 (1999), 3-48 | MR 1673235 | Zbl 0930.60096

[2] K.B. Athreya, P.E. Ney, Branching Processes, Dover Publications Inc. (2004) | MR 2047480 | Zbl 1070.60001 | Zbl 0259.60002

[3] J. Bertoin, Two solvable systems of coagulation equations with limited aggregations, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 no. 6 (2009), 2073-2089 | Numdam | MR 2569886 | Zbl 1179.82180

[4] J. Bertoin, V. Sidoravicius, The structure of typical clusters in large sparse random configurations, J. Stat. Phys. 135 no. 1 (2009), 87-105 | MR 2505727 | Zbl 1168.82028

[5] M. Deaconu, E. Tanré, Smoluchowskiʼs coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 29 no. 3 (2000), 549-579 | Numdam | MR 1817709 | Zbl 1072.60071

[6] P.B. Dubovskiĭ, I.W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci. 19 no. 7 (1996), 571-591 | MR 1385155 | Zbl 0852.45016

[7] M. Dwass, The total progeny in a branching process and a related random walk, J. Appl. Probab. 6 (1969), 682-686 | MR 253433 | Zbl 0192.54401

[8] M.H. Ernst, E.M. Hendriks, R.M. Ziff, Kinetics of gelation and universality, J. Phys. A 16 no. 10 (1983), 2293-2320 | MR 713190

[9] M.H. Ernst, E.M. Hendriks, R.M. Ziff, Coagulation processes with a phase transition, J. Coll. Interface Sci. 97 (1984), 266-277

[10] M. Escobedo, P. Laurençot, S. Mischler, B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations 195 no. 1 (2003), 143-174 | MR 2019246 | Zbl 1133.82316

[11] M. Escobedo, S. Mischler, B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 no. 1 (2002), 157-188 | MR 1947695 | Zbl 1016.82027

[12] N. Fournier, P. Laurençot, Well-posedness of Smoluchowskiʼs coagulation equation for a class of homogeneous kernels, J. Funct. Anal. 233 no. 2 (2006), 351-379 | MR 2214581 | Zbl 1106.45003

[13] N. Fournier, P. Laurençot, Marcus–Lushnikov processes, Smoluchowskiʼs and Floryʼs models, Stochastic Process. Appl. 119 no. 1 (2009), 167-189 | MR 2485023 | Zbl 1169.60027

[14] I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations, Comm. Math. Phys. 194 no. 3 (1998), 541-567 | MR 1631473 | Zbl 0910.60083

[15] N.J. Kokholm, On Smoluchowskiʼs coagulation equation, J. Phys. A 21 no. 3 (1988), 839-842 | MR 930840 | Zbl 0649.34007

[16] P. Laurençot, Global solutions to the discrete coagulation equations, Mathematika 46 no. 2 (1999), 433-442 | MR 1832634 | Zbl 1131.82025

[17] P. Laurençot, On a class of continuous coagulation-fragmentation equations, J. Differential Equations 167 no. 2 (2000), 245-274 | MR 1793195 | Zbl 0978.35083

[18] P. Laurençot, S. Mischler, On coalescence equations and related models, P. Degond, L. Pareschi, G. Russo (ed.), Modeling and Computational Methods for Kinetic Equations, Birkhäuser (2004), 321-356 | MR 2068589 | Zbl 1105.82027

[19] F. Leyvraz, H.R. Tschudi, Singularities in the kinetics of coagulation processes, J. Phys. A 14 no. 12 (1981), 3389-3405 | MR 639565 | Zbl 0481.92020

[20] J.B. Mcleod, On an infinite set of non-linear differential equations, Q. J. Math. Oxford Ser. (2) 13 (1962), 119-128 | MR 139822 | Zbl 0109.31501

[21] G. Menon, R.L. Pego, Approach to self-similarity in Smoluchowskiʼs coagulation equations, Comm. Pure Appl. Math. 57 no. 9 (2004), 1197-1232 | MR 2059679 | Zbl 1049.35048

[22] R. Normand, A model for coagulation with mating, J. Stat. Phys. 137 no. 2 (2009), 343-371 | MR 2559434 | Zbl 1181.82047

[23] J.R. Norris, Smoluchowskiʼs coagulation equation: uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab. 9 no. 1 (1999), 78-109 | MR 1682596 | Zbl 0944.60082

[24] J.R. Norris, Cluster coagulation, Comm. Math. Phys. 209 no. 2 (2000), 407-435 | MR 1737990 | Zbl 0953.60095

[25] G. Stell, R. Ziff, Kinetics of polymer gelation, J. Chem. Phys. 73 (1980), 3492-3499

[26] R. Van Der Hofstad, Random graphs and complex networks, http://www.win.tue.nl/~rhofstad/NotesRGCN2010.pdf | Zbl 06653786

[27] H.J. Van Roessel, M. Shirvani, Some results on the coagulation equation, Nonlinear Anal. 43 no. 5 (2001), 563-573 | MR 1804857 | Zbl 0974.45007

[28] H.J. Van Roessel, M. Shirvani, A formula for the post-gelation mass of a coagulation equation with a separable bilinear kernel, Phys. D 222 no. 1–2 (2006), 29-36 | MR 2265765 | Zbl 1129.82029

[29] M. Von Smoluchowski, Drei vortrage über diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen, Phys. Z 17 (1916), 557-571

[30] H.S. Wilf, Generatingfunctionology, Academic Press (1994), http://www.math.upenn.edu/~wilf/gfology2.pdf | MR 1277813 | Zbl 0831.05001

[31] R.M. Ziff, Kinetics of polymerization, J. Stat. Phys. 23 no. 2 (1980), 241-263 | MR 586509