August 2018, 11(4): 933-952. doi: 10.3934/krm.2018037

Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

1. 

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, 118 route de Narbonne, F-31062 Toulouse cedex 9, France

2. 

Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received  May 2017 Revised  December 2017 Published  April 2018

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $γ < 1$. Due to the property of the diagonal kernel, the value of a solution at a given cluster size depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.

Citation: Philippe Laurençot, Barbara Niethammer, Juan J.L. Velázquez. Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel. Kinetic & Related Models, 2018, 11 (4) : 933-952. doi: 10.3934/krm.2018037
References:
[1]

D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3-48. doi: 10.2307/3318611.

[2]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, arXiv: 1609.03846, 2016.

[3]

E. Buffet and J. V. Pulé, Gelation: the diagonal case revisited, Nonlinearity, 2 (1989), 373-381. doi: 10.1088/0951-7715/2/2/011.

[4]

J. Dolbeault and M. Escobedo, $L^1$ and $L^∞$ intermediate asymptotics for scalar conservation laws, Asymptot. Anal., 41 (2005), 189-213.

[5]

R. -L. Drake, A general mathematical survey of the coagulation equation, In Topics in Current Aerosol Research (part 2), Hidy G. M., Brock, J. R. eds., International Reviews in Aerosol Physics and Chemistry, 203–376, Pergamon Press, Oxford, 1972.

[6]

M. EscobedoS. Mischler and M. Rodriguez-Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001.

[7]

F. Filbet and Ph. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132.

[8]

N. Fournier and Ph. Laurençot, Existence of self-similar solutions to Smoluchowski's coagulation equation, Comm. Math. Phys., 256 (2005), 589-609. doi: 10.1007/s00220-004-1258-5.

[9]

S. K. Friedlander and C. S. Wang, The self-preserving particle size distribution for coagulation by Brownian motion, J. Colloid Interface Sci., 22 (1966), 126-132.

[10]

M. HerrmannB. Niethammer and J. J. L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math., 75 (2017), 105-130. doi: 10.1090/qam/1454.

[11]

D. S. Krivitsky, Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function, J. Phys. A: Math. Gen., 28 (1995), 2025-2039. doi: 10.1088/0305-4470/28/7/022.

[12]

Ph. Laurençot, Weak compactness techniques and coagulation equations, In Evolutionary equations with applications in natural sciences, J. Banasiak, M. Mokhtar-Kharroubi (eds.), 199–253, Lecture Notes in Math., 2126, Springer, Cham, 2015. doi: 10.1007/978-3-319-11322-7_5.

[13]

Ph. Laurençot and S. Mischler, On coalescence equations and related models, In Modeling and computational methods for kinetic equations, P. Degond, L. Pareschi, G. Russo (eds.), 321–356, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004.

[14]

M.-H. Lee, A survey of numerical solutions to the coagulation equation, J. Phys. A, 34 (2001), 10219-10241. doi: 10.1088/0305-4470/34/47/323.

[15]

F. Leyvraz, Existence and properties of post-gel solutions of the equations of coagulation, J. Phys. A, 16 (1983), 2861-2873. doi: 10.1088/0305-4470/16/12/032.

[16]

F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep., 383 (2003), 95-212. doi: 10.1016/S0370-1573(03)00241-2.

[17]

F. Leyvraz, Rigorous results in the scaling theory of irreversible aggregation kinetics, J. Nonlinear Math. Phys., 12 (2005), 449-465. doi: 10.2991/jnmp.2005.12.s1.37.

[18]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Comm. Pure Appl. Math., 57 (2004), 1197-1232. doi: 10.1002/cpa.3048.

[19]

B. NiethammerS. Throm and J. J. L. Velázquez, Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1223-1257. doi: 10.1016/j.anihpc.2015.04.002.

[20]

B. Niethammer and J. J. L. Velázquez, Self-similar solutions with fat tails for a coagulation equation with diagonal kernel, C. R. Math. Acad. Sci. Paris, 349 (2011), 559-562. doi: 10.1016/j.crma.2011.03.017.

[21]

B. Niethammer and J. J. L. Velázquez, Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels, Comm. Math. Phys., 318 (2013), 505-532. doi: 10.1007/s00220-012-1553-5.

[22]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, 2000.

[23]

P. G. J. van Dongen and M. H. Ernst, Scaling solutions of Smoluchowski's coagulation equation, J. Statist. Phys., 50 (1988), 295-329. doi: 10.1007/BF01022996.

[24]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

References:
[1]

D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli, 5 (1999), 3-48. doi: 10.2307/3318611.

[2]

E. Bernard, M. Doumic and P. Gabriel, Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts, arXiv: 1609.03846, 2016.

[3]

E. Buffet and J. V. Pulé, Gelation: the diagonal case revisited, Nonlinearity, 2 (1989), 373-381. doi: 10.1088/0951-7715/2/2/011.

[4]

J. Dolbeault and M. Escobedo, $L^1$ and $L^∞$ intermediate asymptotics for scalar conservation laws, Asymptot. Anal., 41 (2005), 189-213.

[5]

R. -L. Drake, A general mathematical survey of the coagulation equation, In Topics in Current Aerosol Research (part 2), Hidy G. M., Brock, J. R. eds., International Reviews in Aerosol Physics and Chemistry, 203–376, Pergamon Press, Oxford, 1972.

[6]

M. EscobedoS. Mischler and M. Rodriguez-Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001.

[7]

F. Filbet and Ph. Laurençot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132.

[8]

N. Fournier and Ph. Laurençot, Existence of self-similar solutions to Smoluchowski's coagulation equation, Comm. Math. Phys., 256 (2005), 589-609. doi: 10.1007/s00220-004-1258-5.

[9]

S. K. Friedlander and C. S. Wang, The self-preserving particle size distribution for coagulation by Brownian motion, J. Colloid Interface Sci., 22 (1966), 126-132.

[10]

M. HerrmannB. Niethammer and J. J. L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math., 75 (2017), 105-130. doi: 10.1090/qam/1454.

[11]

D. S. Krivitsky, Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function, J. Phys. A: Math. Gen., 28 (1995), 2025-2039. doi: 10.1088/0305-4470/28/7/022.

[12]

Ph. Laurençot, Weak compactness techniques and coagulation equations, In Evolutionary equations with applications in natural sciences, J. Banasiak, M. Mokhtar-Kharroubi (eds.), 199–253, Lecture Notes in Math., 2126, Springer, Cham, 2015. doi: 10.1007/978-3-319-11322-7_5.

[13]

Ph. Laurençot and S. Mischler, On coalescence equations and related models, In Modeling and computational methods for kinetic equations, P. Degond, L. Pareschi, G. Russo (eds.), 321–356, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004.

[14]

M.-H. Lee, A survey of numerical solutions to the coagulation equation, J. Phys. A, 34 (2001), 10219-10241. doi: 10.1088/0305-4470/34/47/323.

[15]

F. Leyvraz, Existence and properties of post-gel solutions of the equations of coagulation, J. Phys. A, 16 (1983), 2861-2873. doi: 10.1088/0305-4470/16/12/032.

[16]

F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep., 383 (2003), 95-212. doi: 10.1016/S0370-1573(03)00241-2.

[17]

F. Leyvraz, Rigorous results in the scaling theory of irreversible aggregation kinetics, J. Nonlinear Math. Phys., 12 (2005), 449-465. doi: 10.2991/jnmp.2005.12.s1.37.

[18]

G. Menon and R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Comm. Pure Appl. Math., 57 (2004), 1197-1232. doi: 10.1002/cpa.3048.

[19]

B. NiethammerS. Throm and J. J. L. Velázquez, Self-similar solutions with fat tails for Smoluchowski's coagulation equation with singular kernels, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1223-1257. doi: 10.1016/j.anihpc.2015.04.002.

[20]

B. Niethammer and J. J. L. Velázquez, Self-similar solutions with fat tails for a coagulation equation with diagonal kernel, C. R. Math. Acad. Sci. Paris, 349 (2011), 559-562. doi: 10.1016/j.crma.2011.03.017.

[21]

B. Niethammer and J. J. L. Velázquez, Self-similar solutions with fat tails for Smoluchowski's coagulation equation with locally bounded kernels, Comm. Math. Phys., 318 (2013), 505-532. doi: 10.1007/s00220-012-1553-5.

[22]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, 2000.

[23]

P. G. J. van Dongen and M. H. Ernst, Scaling solutions of Smoluchowski's coagulation equation, J. Statist. Phys., 50 (1988), 295-329. doi: 10.1007/BF01022996.

[24]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

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