On self-similarity and stationary problem for fragmentation and coagulation models

Escobedo, M.; Mischler, S.; Rodriguez Ricard, M.

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 1, pp. 99-125.

[1] Aldous D.J., Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists, Bernoulli 5 (1999) 3-48. | MR | Zbl

[2] Amann H., Ordinary Differential Equations. An Introduction to Nonlinear Analysis, Translated from the German by Gerhard Metzen, de Gruyter Studies in Mathematics, vol. 13, Walter de Gruyter, Berlin, 1990. | MR | Zbl

[3] Arlotti L., Banasiak J., Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss, J. Math. Anal. Appl. 293 (2) (2004) 693-720. | MR | Zbl

[4] Balabane M., Systèmes différentiels, cours de l'Ecole Nationale des Ponts et Chaussées, 1982.

[5] N. Ben Abdallah, M. Escobedo, S. Mischler, Convergence to the equilibrium for the Pauli equation without detailed balance condition, in preparation. | MR

[6] Bertoin J., On small masses in self-similar fragmentations, Stochastic Process. Appl. 109 (1) (2004) 13-22. | MR | Zbl

[7] Bertoin J., The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc. (JEMS) 5 (4) (2003) 395-416. | MR | Zbl

[8] Bertoin J., Eternal solutions to Smoluchowski's coagulation equation with additive kernel and their probabilistic interpretations, Ann. Appl. Probab. 12 (2) (2002) 547-564. | MR | Zbl

[9] Bertoin J., Self-similar fragmentations, Ann. Inst. H. Poincaré Probab. Statist. 38 (3) (2002) 319-340. | Numdam | MR | Zbl

[10] Bertoin J., Homogeneous fragmentation processes, Probab. Theory Related Fields 121 (3) (2001) 301-318. | MR | Zbl

[11] Cazenave T., Haraux A., An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press, Oxford University Press, New York, 1998. | MR | Zbl

[12] Da Costa F.P., On the dynamic scaling behavior of solutions to the discrete Smoluchowski equations, Proc. Edinburgh Math. Soc. 39 (2) (1996) 547-559. | MR | Zbl

[13] Deaconu M., Tanré E., Smoluchowski's coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (4) (2000) 549-579. | Numdam | MR | Zbl

[14] Derrida B., Godréche C., Yekuitieli I., Scale-invariant regimes in one-dimensional models of growing and coalescing droplets, Phys. Rev. A 44 (1991) 6241-6251.

[15] Diperna R.J., Lions P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 707-741. | MR | Zbl

[16] Van Dongen P.G.J., Ernst M.H., Cluster size distribution in irreversible aggregation at large times, J. Phys. A 18 (1985) 2779-2793. | MR

[17] Van Dongen P.G.J., Ernst M.H., Scaling solutions of Smoluchowski's coagulation equation, J. Statist. Phys. 50 (1988) 295-329. | MR | Zbl

[18] Drake R.L., A general mathematical survey of the coagulation equation, in: Topics in Current Aerosol Research (part 2), International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, 1972, pp. 203-376.

[19] Dubovskiĭ P.B., Stewart I.W., Trend to equilibrium for the coagulation-fragmentation equation, Math. Methods Appl. Sci. 19 (1996) 761-772. | MR | Zbl

[20] Edwards R.E., Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, 1965. | MR | Zbl

[21] Eibeck A., Wagner W., Stochastic particle approximations for Smoluchowski's coagulation equation, Ann. Appl. Probab. 11 (2001) 1137-1165. | MR | Zbl

[22] Escobedo M., Laurençot Ph., Mischler S., Perthame B., Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations 195 (1) (2003) 143-174. | MR | Zbl

[23] Escobedo M., Mischler S., Perthame B., Gelation in coagulation and fragmentation models, Comm. Math. Phys. 231 (2002) 157-188. | MR | Zbl

[24] Escobedo M., Zuazua E., Large time behavior of the solutions of a convection diffusion equation, J. Funct. Anal. 100 (1991) 119-161. | MR | Zbl

[25] Fournier N., Giet J.-S., On small particles in coagulation-fragmentation equations, J. Statist. Phys. 111 (5-6) (2003) 1299-1329. | MR | Zbl

[26] N. Fournier, S. Mischler, Trend to the equilibrium for the coagulation equation with strong fragmentation but with balance condition, in: Proceedings: Mathematical, Physical and Engineering Sciences, in press. | Zbl

[27] N. Fournier, S. Mischler, On a Boltzmann equation for elastic, inelastic and coalescing collisions, preprint, 2003, submitted for publication.

[28] Gamba I.M., Panferov V., Villani C., On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246 (3) (2004) 503-541. | MR | Zbl

[29] Haas B., Loss of mass in deterministic and random fragmentations, Stochastic Process. Appl. 106 (2) (2003) 245-277. | MR | Zbl

[30] Kreer M., Penrose O., Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel, J. Statist. Phys. 75 (1994) 389-407. | MR | Zbl

[31] Krivitsky D.S., Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function, J. Phys. A 28 (1995) 2025-2039. | MR | Zbl

[32] Ph. Laurençot, Convergence to self-similar solutions for coagulation equation, preprint, 2003.

[33] Laurençot Ph., Mischler S., From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A 132 (5) (2002) 1219-1248. | MR | Zbl

[34] Laurençot Ph., Mischler S., The continuous coagulation-fragmentation equations with diffusion, Arch. Rational Mech. Anal. 162 (2002) 45-99. | MR | Zbl

[35] Laurençot Ph., Mischler S., Convergence to equilibrium for the continuous coagulation-fragmentation equation, Bull. Sci. Math. 127 (2003) 179-190. | MR | Zbl

[36] Laurençot P., Mischler S., On coalescence equations and related models, in: Degond P., Pareschi L., Russo G. (Eds.), Modelling and Computational Methods for Kinetic Equations, Series Modelling and Simulation in Science, Engineering and Technology (MSSET), Birkhäuser, 2004, submitted for publication. | MR | Zbl

[37] Leyvraz F., Existence and properties of post-gel solutions for the kinetic equations of coagulation, J. Phys. A 16 (1983) 2861-2873. | MR

[38] Leyvraz F., Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Reports 383 (2-3) (2003) 95-212.

[39] G. Menon, R.L. Pego, Approach to self-similarity in Smoluchowski's coagulation equation, preprint, 2003.

[40] G. Menon, R.L. Pego, Dynamical scaling in Smoluchowski's coagulation equation: uniform convergence, preprint, 2003.

[41] S. Mischler, C. Mouhot, M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations, in preparation.

[42] Mischler S., Rodriguez Ricard M., Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003) 407-412. | MR | Zbl

[43] Mischler S., Wennberg B., On the spatially homogeneous Boltzmann equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (4) (1999) 467-501. | Numdam | MR | Zbl

[44] Tanaka H., Inaba S., Nakaza K., Steady-state size distribution for self-similar collision cascade, Icarus 123 (1996) 450-455.