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December  2020, 13(12): 3319-3334. doi: 10.3934/dcdss.2020161

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

Jacek Banasiak 1,2,

1. 

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa
2. 

Institute of Mathematics, Łódź University of Technology, Łódź, Poland

The paper is dedicated to Giséle Ruiz Goldstein on the occasion of her birthday

Received  February 2019 Revised  April 2019 Published  December 2020 Early access  December 2019

Fund Project: The research has been partially supported by the National Science Centre of Poland Grant 2017/25/B/ST1/00051 and the National Research Foundation of South Africa Grant 82770

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

Keywords: Coagulation-fragmentation equations, analytic semigroups, moment estimates, interpolation spaces.
Mathematics Subject Classification: Primary: 35R09, 47J35; Secondary: 35K58, 47D06, 82C22, 82C05.
Citation: Jacek Banasiak. Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3319-3334. doi: 10.3934/dcdss.2020161
References:
[1]

M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys., 65 (1079), 203-230.  doi: 10.1007/BF01197880.  Google Scholar

[2]

W. Arendt and A. Rhandi, Perturbation of positive semigroups, Arch. Math. (Basel), 56 (1991), 107-119.  doi: 10.1007/BF01200341.  Google Scholar

[3]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Statist. Phys., 61 (1990), 203-234.  doi: 10.1007/BF01013961.  Google Scholar

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[5] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, Volume I & II, Chapman & Hall/CRC Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, 2019.   Google Scholar
[6]

J. Banasiak, L. O. Joel and S. Shindin, The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation, Kinetic and Related Models, 12 (2019), 1069–1092, arXiv: 1809.00046. Google Scholar

[7]

J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, J. Math. Anal. Appl., 391 (2012), 312-322.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[8]

J. Banasiak, W. Lamb and M. Langer, Strong fragmentation and coagulation with power-law rates, J. Engrg. Math., 82 (2013), 199-215.  doi: 10.1007/s10665-012-9596-3.  Google Scholar

[9]

R. Becker and W. Döring, Kinetische behandlung der keimbildung in übersättigten dämpfen, Annalen der Physik, 416 (1935), 719-752.   Google Scholar

[10]

J. Bergh and J. Löfström, Interpolation Spaces: An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[11] J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511617768.  Google Scholar
[12]

P. J. Blatz and A. V. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, The Journal of Physical Chemistry, 49 (1945), 77-80.  doi: 10.1021/j150440a004.  Google Scholar

[13]

P. B. Dubovskiǐ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci., 19 (1996), 571-591.  doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q.  Google Scholar

[14]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.  Google Scholar

[15]

M. Escobedo, S. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231 (2002), 157-188.  doi: 10.1007/s00220-002-0680-9.  Google Scholar

[16]

M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.  Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[18]

P. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219-1248.  doi: 10.1017/S0308210502000598.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[20]

E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895.  doi: 10.1103/PhysRevLett.58.892.  Google Scholar

[21]

Z. A. Melzak, A scalar transport equation, Trans. Amer. Math. Soc., 85 (1957), 547-560.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[22]

H. Müller, Zur allgemeinen theorie der raschen koagulation, Fortschrittsberichte über Kolloide und Polymere, 27 (1928), 223–250. Google Scholar

[23]

M. v. Smoluchowski, Drei vortrage über diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift für Physik, 17 (1916), 557–585. Google Scholar

[24]

M. v. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift für Physikalische Chemie, 92 (1917), 129–168. doi: 10.1515/zpch-1918-9209.  Google Scholar

[25]

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.  Google Scholar

[26]

I. W. Stewart, Density conservation for a coagulation equation, Z. Angew. Math. Phys., 42 (1991), 746-756.  doi: 10.1007/BF00944770.  Google Scholar

[27]

R. D. Vigil and R. M. Ziff, On the scaling theory of two-component aggregation, Chemical Engineering Science, 53 (1998), 1725-1729.  doi: 10.1016/S0009-2509(98)00016-5.  Google Scholar

[28]

J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171.  doi: 10.1007/BF01351602.  Google Scholar

show all references

References:
[1]

M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers, Comm. Math. Phys., 65 (1079), 203-230.  doi: 10.1007/BF01197880.  Google Scholar

[2]

W. Arendt and A. Rhandi, Perturbation of positive semigroups, Arch. Math. (Basel), 56 (1991), 107-119.  doi: 10.1007/BF01200341.  Google Scholar

[3]

J. M. Ball and J. Carr, The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation, J. Statist. Phys., 61 (1990), 203-234.  doi: 10.1007/BF01013961.  Google Scholar

[4]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2006.  Google Scholar

[5] J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, Volume I & II, Chapman & Hall/CRC Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, 2019.   Google Scholar
[6]

J. Banasiak, L. O. Joel and S. Shindin, The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation, Kinetic and Related Models, 12 (2019), 1069–1092, arXiv: 1809.00046. Google Scholar

[7]

J. Banasiak and W. Lamb, Analytic fragmentation semigroups and continuous coagulation-fragmentation equations with unbounded rates, J. Math. Anal. Appl., 391 (2012), 312-322.  doi: 10.1016/j.jmaa.2012.02.002.  Google Scholar

[8]

J. Banasiak, W. Lamb and M. Langer, Strong fragmentation and coagulation with power-law rates, J. Engrg. Math., 82 (2013), 199-215.  doi: 10.1007/s10665-012-9596-3.  Google Scholar

[9]

R. Becker and W. Döring, Kinetische behandlung der keimbildung in übersättigten dämpfen, Annalen der Physik, 416 (1935), 719-752.   Google Scholar

[10]

J. Bergh and J. Löfström, Interpolation Spaces: An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[11] J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006.  doi: 10.1017/CBO9780511617768.  Google Scholar
[12]

P. J. Blatz and A. V. Tobolsky, Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, The Journal of Physical Chemistry, 49 (1945), 77-80.  doi: 10.1021/j150440a004.  Google Scholar

[13]

P. B. Dubovskiǐ and I. W. Stewart, Existence, uniqueness and mass conservation for the coagulation-fragmentation equation, Math. Methods Appl. Sci., 19 (1996), 571-591.  doi: 10.1002/(SICI)1099-1476(19960510)19:7<571::AID-MMA790>3.0.CO;2-Q.  Google Scholar

[14]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.  Google Scholar

[15]

M. Escobedo, S. Mischler and B. Perthame, Gelation in coagulation and fragmentation models, Comm. Math. Phys., 231 (2002), 157-188.  doi: 10.1007/s00220-002-0680-9.  Google Scholar

[16]

M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models, J. Differential Equations, 195 (2003), 143-174.  doi: 10.1016/S0022-0396(03)00134-7.  Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[18]

P. Laurençot and S. Mischler, From the discrete to the continuous coagulation-fragmentation equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1219-1248.  doi: 10.1017/S0308210502000598.  Google Scholar

[19]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.  Google Scholar

[20]

E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895.  doi: 10.1103/PhysRevLett.58.892.  Google Scholar

[21]

Z. A. Melzak, A scalar transport equation, Trans. Amer. Math. Soc., 85 (1957), 547-560.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[22]

H. Müller, Zur allgemeinen theorie der raschen koagulation, Fortschrittsberichte über Kolloide und Polymere, 27 (1928), 223–250. Google Scholar

[23]

M. v. Smoluchowski, Drei vortrage über diffusion, brownsche bewegung und koagulation von kolloidteilchen, Zeitschrift für Physik, 17 (1916), 557–585. Google Scholar

[24]

M. v. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen, Zeitschrift für Physikalische Chemie, 92 (1917), 129–168. doi: 10.1515/zpch-1918-9209.  Google Scholar

[25]

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.  Google Scholar

[26]

I. W. Stewart, Density conservation for a coagulation equation, Z. Angew. Math. Phys., 42 (1991), 746-756.  doi: 10.1007/BF00944770.  Google Scholar

[27]

R. D. Vigil and R. M. Ziff, On the scaling theory of two-component aggregation, Chemical Engineering Science, 53 (1998), 1725-1729.  doi: 10.1016/S0009-2509(98)00016-5.  Google Scholar

[28]

J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., 229 (1977), 163-171.  doi: 10.1007/BF01351602.  Google Scholar

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