In this paper we construct classical solutions of a family of coagulation equations with homogeneous kernels that exhibit the behaviour known as gelation. This behaviour consists in the loss of mass due to the fact that some of the particles can become infinitely large in finite time.
@article{AIHPC_2012__29_4_589_0, author = {Escobedo, M. and Vel\'azquez, J.J.L.}, title = {Classical non-mass-preserving solutions of coagulation equations}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {29}, year = {2012}, pages = {589-635}, doi = {10.1016/j.anihpc.2012.03.001}, mrnumber = {2948290}, zbl = {1251.35082}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_4_589_0} }
Escobedo, M.; Velázquez, J.J.L. Classical non-mass-preserving solutions of coagulation equations. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 589-635. doi : 10.1016/j.anihpc.2012.03.001. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_4_589_0/
[1] Post-gelation solutions to Smoluchowskiʼs coagulation equation, J. Phys. A 27 no. 12 (1994), 4203-4209 | MR 1282618 | Zbl 0827.60094
, ,[2] Stability of weak-turbulence Kolmogorov spectra, (ed.), Nonlinear Waves and Weak Turbulence, AMS Translations Series 2 vol. 182 (1998), 1-81 | MR 1618499 | Zbl 0898.76047
, ,[3] Coagulation processes with a phase transition, J. Colloid Interface Sci. 97 (1984), 266-277
, , ,[4] Gelation and mass conservation in coagulation–fragmentation models, J. Differential Equations 195 no. 1 (2003), 143-174 | MR 2019246 | Zbl 1133.82316
, , , ,[5] Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 no. 1 (2002), 157-188 | MR 1947695 | Zbl 1016.82027
, , ,[6] On the fundamental solution of the linearized Uehling–Uhlenbeck equation, Arch. Ration. Mech. Anal. 186 (2007), 309-349 | MR 2342205 | Zbl 05211791
, , ,[7] Singular solutions for the Uehling–Uhlenbeck equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 67-107 | MR 2388938 | Zbl 1148.35015
, , ,[8] On the fundamental solution of a homogeneous linearized coagulation equation, Comm. Math. Phys. 3 no. 297 (2010), 759-816 | MR 2653902 | Zbl 1193.82061
, ,[9] M. Escobedo, J.J.L. Velázquez, Local well posedness for a linear coagulation equation, Trans. Amer. Math. Soc., in press. | MR 3009645
[10] Marcus–Lushnikov processes, Smoluchowskiʼs and Floryʼs models, Stoch. Process. Appl. 119 (2009), 167-189 | MR 2485023 | Zbl 1169.60027
, ,[11] Molecular size distribution in three dimensional polymers. II. Trifunctional branching units, J. Amer. Chem. Soc. 63 (1941), 3091-3096
,[12] Existence of gelling solutions for coagulation–fragmentation equations, Comm. Math. Phys. 194 (1998), 541-567 | MR 1631473 | Zbl 0910.60083
,[13] Dynamical formation of a Bose–Einstein condensate, Physica D 152–153 (2001), 779-786 | MR 1837939 | Zbl 0979.83028
, , , ,[14] On a class of continuous coagulation–fragmentation equations, J. Differential Equations 167 (2000), 245-274 | MR 1793195 | Zbl 0978.35083
,[15] A modified Boltzmann equation for Bose–Einstein particles: isotropic solutions and long time behavior, J. Stat. Phys. 98 (2000), 1335-1394 | MR 1751703 | Zbl 1005.82026
,[16] On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles, J. Stat. Phys. 116 (2004), 1597-1649 | MR 2096049 | Zbl 1097.82023
,[17] On the scalar transport equation, Proc. London Math. Soc. 14 no. 3 (1964), 445-458 | MR 162110 | Zbl 0123.29901
,[18] Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin (1996) | MR 1419319 | Zbl 0873.35001
, ,[19] Kinetics of Bose condensation, Phys. Rev. Lett. 74 (1995), 3093-3097
, ,[20] Condensation of bosons in the kinetic regime, Phys. Rev. D 55 no. 2 (1997), 489-502
, ,[21] Kinetics of the Bose–Einstein condensation, Physica D 239 (2010), 627-634 | MR 2601928 | Zbl 1186.82052
,[22] Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys. 11 no. 2 (1943), 45-55
,[23] Steady-state size distribution for the self-similar collision cascade, Icarus 123 (1996), 450-455
, , ,