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

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

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.

Citation: Jacek Banasiak. Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients. Discrete and 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.

[2]

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

[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.

[4]

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

[5] J. BanasiakW. 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. 
[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.

[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.

[8]

J. BanasiakW. 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.

[9]

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

[10]

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

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

[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.

[14]

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

[15]

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

[16]

M. EscobedoP. LaurençotS. 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.

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

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.

[2]

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

[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.

[4]

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

[5] J. BanasiakW. 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. 
[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.

[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.

[8]

J. BanasiakW. 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.

[9]

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

[10]

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

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

[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.

[14]

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

[15]

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

[16]

M. EscobedoP. LaurençotS. 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.

[17]

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

[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.

[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.

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

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