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June  2011, 4(2): 427-439. doi: 10.3934/krm.2011.4.427

## On a model for mass aggregation with maximal size

 1 Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands 2 Department of Mathematics, Saarland University, 66123 Saarbrücken, Germany 3 Oxford Centre of Nonlinear PDE, Mathematical Insitute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, United Kingdom 4 School of Computer and Communication Sciences, École polytechnique fédérale de Lausanne, CH - 1015 Lausanne, Switzerland

Received  October 2010 Revised  December 2010 Published  April 2011

We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study the existence of self-similar solutions, and analyze the large-time behavior mostly by numerical simulations. Depending on the parameter $k_0$, which controls the probability of coagulation, we observe two different scenarios: For $k_0>2$ there exist two self-similar solutions to the mean field equation, of which one is unstable. In numerical simulations we observe that for all initial data the rescaled solutions converge to the stable self-similar solution. For $k_0<2$, however, no self-similar behavior occurs as the solutions converge in the original variables to a limit that depends strongly on the initial data. We prove rigorously a corresponding statement for $k_0\in (0,1/3)$. Simulations for the cross-over case $k_0=2$ are not completely conclusive, but indicate that, depending on the initial data, part of the mass evolves in a self-similar fashion whereas another part of the mass remains in the small particles.
Citation: Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
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##### References:
 [1] Phys. Rev. E, 76 (2007), 060102. doi: 10.1103/PhysRevE.76.060102.  Google Scholar [2] In G. M. Hidy and J. R. Brock eds., "Topics in current aerosol research (Part 2)"; International reviews in Aerosol Physics and Chemistry, Pergamon (1972), 201-376 Google Scholar [3] Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.  Google Scholar [4] Comm. Math. Phys., 256 (2005) 589-609. doi: 10.1007/s00220-004-1258-5.  Google Scholar [5] J. Funct. Anal., 233 (2006) 351-379. doi: 10.1016/j.jfa.2005.07.013.  Google Scholar [6] Wiley, New York, 1977. Google Scholar [7] J. Nonlinear Science, 13 (2003), 311-346. doi: 10.1007/s00332-002-0543-8.  Google Scholar [8] Phys. Reports, 383 (2003), 95-212. doi: 10.1016/S0370-1573(03)00241-2.  Google Scholar [9] Comm. Pure Appl. Math., 57 (2004), 1197-1232. doi: 10.1002/cpa.3048.  Google Scholar [10] Trans. AMS, 362 (2010), 6551-6590. doi: 10.1090/S0002-9947-2010-05085-8.  Google Scholar [11] Phys. Zeitschr., 17 (1916), 557-599. Google Scholar [12] J. Statist. Phys., 23 (1980), 241-263. doi: 10.1007/BF01012594.  Google Scholar
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