A kinetic equation for repulsive coalescing random jumps in continuum

Krzysztof Pilorz


A continuum individual-based model of hopping and coalescing particles is introduced and studied. Its microscopic dynamics are described by a hierarchy of evolution equations obtained in the paper. Then the passage from the micro- to mesoscopic dynamics is performed by means of a Vlasov-type scaling. The existence and uniqueness of solutions of the corresponding kinetic equation are proved.


Coalescence; coagulation; hopping particles; individual-based model; configuration spaces; infinite particle system; microscopic dynamics; Vlasov scaling; kinetic equation

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

Banasiak, J., Kinetic models in natural sciences, in Evolutionary Equations with Applications in Natural Sciences, volume 2126 of Lecture Notes in Math., 133-198, Springer, Cham, 2015.

Banasiak, J., Lamb, W., Langer, M., Strong fragmentation and coagulation with power-law rates, J. Engrg. Math. 82 (2013), 199-215.

Belavkin, V. P., Kolokoltsov, V. N., On a general kinetic equation for many-particle systems with interaction, fragmentation and coagulation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2031) (2003), 727-748.

Berns, C., Kondratiev, Y., Kozitsky, Y., Kutoviy, O., Kawasaki dynamics in continuum: micro- and mesoscopic descriptions, J. Dynam. Differential Equations 25 (4) (2013), 1027-1056.

Capitan, J. A., Delius, G. W., Scale-invariant model of marine population dynamics, Phys. Rev. E (3) 81 (6) (2010), 061901, 15pp.

Finkelshtein, D., Kondratiev, Y., Kozitsky, Y., Kutoviy, O., The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci. 25 (2) (2015), 343-370.

Finkelshtein, D., Kondratiev, Y., Kutoviy, O., Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys. 141 (1) (2010), 158-178.

Finkelshtein, D., Kondratiev, Y., Kutoviy, O., Statistical dynamics of continuous systems: perturbative and approximative approaches, Arabian Journal of Mathematics 4 (4) (2015), 255-300.

Finkelshtein, D., Kondratiev, Y., Joao Oliveira, M., Markov evolutions and hierarchical equations in the continuum. I. One-component systems, J. Evol. Equ. 9 (2) (2009), 197-233.

Kolokoltsov, V. N., Hydrodynamic limit of coagulation-fragmentation type models of k-nary interacting particles, J. Statist. Phys. 115 (5-6) (2004), 1621-1653.

Kondratiev, Y., Kuna, T., Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2) (2002), 201-233.

Lachowicz, M., Laurencot, P., Wrzosek, D., On the Oort-Hulst-Safronov coagulation equation and its relation to the Smoluchowski equation, SIAM J. Math. Anal. 34 (6) (2003), 1399-1421 (electronic).

Lamb, W., Applying functional analytic techniques to evolution equations, in Evolutionary Equations with Applications in Natural Sciences, volume 2126 of Lecture Notes in Math., 1-46, Springer, Cham, 2015.

Rudnicki, R., Wieczorek, R., Fragmentation-coagulation models of phytoplankton, Bull. Pol. Acad. Sci. Math. 54 (2) (2006), 175-191.

Rudnicki, R., Wieczorek, R., Phytoplankton dynamics: from the behaviour of cells to a transport equation, Math. Model. Nat. Phenom. 1 (1) (2006), 83-100.

DOI: http://dx.doi.org/10.17951/a.2016.70.1.47
Data publikacji: 2016-07-04 15:44:01
Data złożenia artykułu: 2016-07-04 12:16:57


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