Advanced Search
Article Contents
Article Contents

Well-posedness for boundary value problems for coagulation-fragmentation equations

Abstract Full Text(HTML) Related Papers Cited by
  • We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular kernels relevant in the applications. We determine the large time asymptotic behavior of solutions, proving that solutions converge exponentially fast to zero in the absence of fragmentation and stabilize toward an equilibrium if the boundary value satisfies a detailed balance condition. Incidentally, we obtain an improvement in the regularity of solutions by showing the finiteness of negative moments for positive time.

    Mathematics Subject Classification: 35A01, 35R09.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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] J. BanasiakW. Lamb and  P. LaurençotAnalytic methods for coagulation-fragmentation models, Chapman and Hall/CRC Press, 2019. 
    [3] V. I. Bogachev, Measure theory, Vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.
    [4] C. C. CamejoR. Gröpler and G. Warnecke, Regular solutions to the coagulation equations with singular kernels, Math. Methods Appl. Sci., 38 (2015), 2171-2184.  doi: 10.1002/mma.3211.
    [5] J. A. Cañizo, Convergence to equilibrium for the discrete coagulation-fragmentation equations with detailed balance, J. Stat. Phys., 129 (2007), 1-26.  doi: 10.1007/s10955-007-9373-2.
    [6] R. L. Drake, A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), 3 (1972), 201-376. 
    [7] R. J. Field and R. M. Noyes, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, The Journal of Chemical Physics, 60 (1974), 1877-1884.  doi: 10.1063/1.1681288.
    [8] S. K. FriedlanderSmoke, dust, and haze, 2$^{nd}$ edition, Oxford University Press, New York, 2000. 
    [9] P. Laurençot and S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Ration. Mech. Anal., 162 (2002), 45-99.  doi: 10.1007/s002050100186.
    [10] M. J. McGrathT. OleniusI. K. OrtegaV. LoukonenP. PaasonenT. KurténM. Kulmala and H. Vehkamäki, Atmospheric cluster dynamics code: A flexible method for solution of the birth-death equations, Atmos. Chem. Phys., 12 (2012), 2345-2355.  doi: 10.5194/acp-12-2345-2012.
    [11] Z. A. Melzak, A scalar transport equation, Trans. Amer. Math. Soc., 85 (1957), 547-560.  doi: 10.1090/S0002-9947-1957-0087880-6.
    [12] J. R. Norris, Smoluchowski's coagulation equation: Uniqueness, nonuniqueness and a hydrodynamic limit for the stochastic coalescent, Ann. Appl. Probab., 9 (1999), 78-109.  doi: 10.1214/aoap/1029962598.
    [13] T. Olenius, O. Kupiainen-Määttä, I. K. Ortega, T. Kurtén and H. Vehkamäki, Free energy barrier in the growth of sulfuric acid–ammonia and sulfuric acid–dimethylamine clusters, J. Chem. Phys., 139 (2013), 084312. doi: 10.1063/1.4819024.
    [14] A. S. Perelson and R. W. Samsel, Kinetics of red blood cell aggregation: An example of geometric polymerization, in Kinetics of Aggregation and Gelation, North Holland, Elsevier, 1984,137–144. doi: 10.1016/B978-0-444-86912-8.50035-3.
    [15] V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations, Jerusalem, 1972.
    [16] J. Saha and J. Kumar, The singular coagulation equation with multiple fragmentation, Z. Angew. Math. Phys., 66 (2015), 919-941.  doi: 10.1007/s00033-014-0452-3.
    [17] M. v. Smoluchowski, Drei Vorträge über Diffusion. Brownsche Bewegung und Koagulation von Kolloidteilchen, Z. Phys., 17 (1916), 557-585. 
    [18] M. v. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift für physikalische Chemie, 92 (1918), 129–168. doi: 10.1515/zpch-1918-9209.
    [19] I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels, Math. Methods Appl. Sci., 11 (1989), 627-648.  doi: 10.1002/mma.1670110505.
    [20] W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys., 11 (1943), 45-55.  doi: 10.1063/1.1723803.
  • 加载中

Article Metrics

HTML views(272) PDF downloads(249) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint