• Previous Article
    Local well-posedness of the Boltzmann equation with polynomially decaying initial data
  • KRM Home
  • This Issue
  • Next Article
    Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime
August  2020, 13(4): 815-835. doi: 10.3934/krm.2020028

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

Universitat Politècnica de Catalunya - BGSMath, Departament de Matemàtiques, Av. Diagonal 647, 08028 Barcelona, Spain

Received  October 2019 Revised  January 2020 Published  May 2020

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.

Citation: Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028
References:
[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.  Google Scholar

[2] J. BanasiakW. Lamb and P. Laurençot, Analytic methods for coagulation-fragmentation models, Chapman and Hall/CRC Press, 2019.   Google Scholar
[3]

V. I. Bogachev, Measure theory, Vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. L. Drake, A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), 3 (1972), 201-376.   Google Scholar

[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.  Google Scholar

[8] S. K. Friedlander, Smoke, dust, and haze, 2$^{nd}$ edition, Oxford University Press, New York, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations, Jerusalem, 1972. Google Scholar

[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.  Google Scholar

[17]

M. v. Smoluchowski, Drei Vorträge über Diffusion. Brownsche Bewegung und Koagulation von Kolloidteilchen, Z. Phys., 17 (1916), 557-585.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2] J. BanasiakW. Lamb and P. Laurençot, Analytic methods for coagulation-fragmentation models, Chapman and Hall/CRC Press, 2019.   Google Scholar
[3]

V. I. Bogachev, Measure theory, Vol. 1, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. L. Drake, A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research (Part 2), 3 (1972), 201-376.   Google Scholar

[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.  Google Scholar

[8] S. K. Friedlander, Smoke, dust, and haze, 2$^{nd}$ edition, Oxford University Press, New York, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

V. S. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations, Jerusalem, 1972. Google Scholar

[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.  Google Scholar

[17]

M. v. Smoluchowski, Drei Vorträge über Diffusion. Brownsche Bewegung und Koagulation von Kolloidteilchen, Z. Phys., 17 (1916), 557-585.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014

[2]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043

[3]

Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009

[4]

Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541

[5]

Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040

[6]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

[7]

Jacek Banasiak. Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020161

[8]

Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669

[9]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041

[10]

Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2020, 9 (2) : 431-446. doi: 10.3934/eect.2020012

[11]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions of a free boundary problem modelling the growth of tumors with Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 625-651. doi: 10.3934/dcds.2009.24.625

[12]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

[13]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197

[14]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011

[15]

John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291

[16]

Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436

[17]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

[18]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651

[19]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683

[20]

Akisato Kubo. Asymptotic behavior of solutions of the mixed problem for semilinear hyperbolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 59-74. doi: 10.3934/cpaa.2004.3.59

2018 Impact Factor: 1.38

Article outline

[Back to Top]