# American Institute of Mathematical Sciences

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. Google Scholar [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.Google Scholar [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. Google Scholar [4] J. Dolbeault and M. Escobedo, $L^1$ and $L^∞$ intermediate asymptotics for scalar conservation laws, Asymptot. Anal., 41 (2005), 189-213. Google Scholar [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.Google Scholar [6] M. Escobedo, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] M. Herrmann, B. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] B. Niethammer, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [22] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, 2000.Google Scholar [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. Google Scholar [24] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar

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##### 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. Google Scholar [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.Google Scholar [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. Google Scholar [4] J. Dolbeault and M. Escobedo, $L^1$ and $L^∞$ intermediate asymptotics for scalar conservation laws, Asymptot. Anal., 41 (2005), 189-213. Google Scholar [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.Google Scholar [6] M. Escobedo, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] M. Herrmann, B. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] B. Niethammer, S. 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. Google Scholar [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. Google Scholar [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. Google Scholar [22] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, 2000.Google Scholar [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. Google Scholar [24] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. Google Scholar
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