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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
References:
[1]

A. Boudaoud, J. Bico and B. Roman, Elastocapillary coalescence: Aggregation and fragmentation with maximal size,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.060102. Google Scholar

[2]

R. L. Drake, A general mathematical survey of the coagulation equation,, In G. M. Hidy and J. R. Brock eds., (1972), 201. Google Scholar

[3]

M. Escobedo, S. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problems for fragmentation and coagulation models,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 99. Google Scholar

[4]

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

[5]

N. Fournier and P. Laurençot, Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels,, J. Funct. Anal., 233 (2006), 351. doi: 10.1016/j.jfa.2005.07.013. Google Scholar

[6]

S. K. Friedlander, "Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics,", Wiley, (1977). Google Scholar

[7]

T. Gallay and A. Mielke, Convergence results for a coarsening model using global linearization,, J. Nonlinear Science, 13 (2003), 311. doi: 10.1007/s00332-002-0543-8. Google Scholar

[8]

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

[9]

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

[10]

G. Menon, B. Niethammer and R. L. Pego, Dynamics and self-similarity in min-driven clustering,, Trans. AMS, 362 (2010), 6551. doi: 10.1090/S0002-9947-2010-05085-8. Google Scholar

[11]

M. Smoluchowski, Drei vorträge über diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen,, Phys. Zeitschr., 17 (1916), 557. Google Scholar

[12]

R. M. Ziff, Kinetics of polymerization,, J. Statist. Phys., 23 (1980), 241. doi: 10.1007/BF01012594. Google Scholar

show all references

References:
[1]

A. Boudaoud, J. Bico and B. Roman, Elastocapillary coalescence: Aggregation and fragmentation with maximal size,, Phys. Rev. E, 76 (2007). doi: 10.1103/PhysRevE.76.060102. Google Scholar

[2]

R. L. Drake, A general mathematical survey of the coagulation equation,, In G. M. Hidy and J. R. Brock eds., (1972), 201. Google Scholar

[3]

M. Escobedo, S. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problems for fragmentation and coagulation models,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 99. Google Scholar

[4]

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

[5]

N. Fournier and P. Laurençot, Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels,, J. Funct. Anal., 233 (2006), 351. doi: 10.1016/j.jfa.2005.07.013. Google Scholar

[6]

S. K. Friedlander, "Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics,", Wiley, (1977). Google Scholar

[7]

T. Gallay and A. Mielke, Convergence results for a coarsening model using global linearization,, J. Nonlinear Science, 13 (2003), 311. doi: 10.1007/s00332-002-0543-8. Google Scholar

[8]

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

[9]

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

[10]

G. Menon, B. Niethammer and R. L. Pego, Dynamics and self-similarity in min-driven clustering,, Trans. AMS, 362 (2010), 6551. doi: 10.1090/S0002-9947-2010-05085-8. Google Scholar

[11]

M. Smoluchowski, Drei vorträge über diffusion, brownsche molekularbewegung und koagulation von kolloidteilchen,, Phys. Zeitschr., 17 (1916), 557. Google Scholar

[12]

R. M. Ziff, Kinetics of polymerization,, J. Statist. Phys., 23 (1980), 241. doi: 10.1007/BF01012594. Google Scholar

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