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January  2018, 23(1): 13-27. doi: 10.3934/dcdsb.2018002

Self-similar solutions of fragmentation equations revisited

Institute of Mathematics, University of Silesia, 40-007 Katowice, Poland

* Corresponding author: Marta Tyran-Kamińska

Received  October 2016 Published  January 2018

Fund Project: This research was supported by the Polish NCN grant No. 2014/13/B/ST1/00224

We study the large time behaviour of the mass (size) of particles described by the fragmentation equation with homogeneous breakup kernel. We give necessary and sufficient conditions for the convergence of solutions to the unique self-similar solution.

Citation: Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
References:
[1]

O. Arino and R. Rudnicki, Stability of phytoplankton dynamics, C. R. Biologies, 327 (2004), 961-969. doi: 10.1016/j.crvi.2004.03.013. Google Scholar

[2]

L. Arlotti and J. Banasiak, Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss, J. Math. Anal. Appl., 293 (2004), 693-720. doi: 10.1016/j.jmaa.2004.01.028. Google Scholar

[3]

J. Banasiak, On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math., 5 (2001), 169-191. doi: 10.11650/twjm/1500574893. Google Scholar

[4]

J. Banasiak, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci., 14 (2004), 483-501. doi: 10.1142/S0218202504003325. Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications Springer-Verlag London Ltd., London, 2006. doi: 10.1007/1-84628-153-9. Google Scholar

[6]

J. Banasiak and W. Lamb, On the application of substochastic semigroup theory to fragmentation models with mass loss, J. Math. Anal. Appl., 284 (2003), 9-30. doi: 10.1016/S0022-247X(03)00154-9. Google Scholar

[7]

J. Banasiak and M. Mokhtar-Kharroubi, Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 529-542. doi: 10.3934/dcdsb.2005.5.529. Google Scholar

[8]

J. Bertoin, Random Fragmentation and Coagulation Processes Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617768. Google Scholar

[9]

J. Bertoin and M.-E. Caballero, Entrance from 0+ for increasing semi-stable Markov processes, Bernoulli, 8 (2002), 195-205. Google Scholar

[10]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61. doi: 10.1017/apr.2016.41. Google Scholar

[11]

W. Biedrzycka and M. Tyran-Kamińska, Existence of invariant densities for semiflows with jumps, J. Math. Anal. Appl., 435 (2016), 61-84. doi: 10.1016/j.jmaa.2015.10.019. Google Scholar

[12]

M. E. Caballero and V. Rivero, On the asymptotic behaviour of increasing self-similar Markov processes, Electron. J. Probab., 14 (2009), 865-894. doi: 10.1214/EJP.v14-637. Google Scholar

[13]

P. Carmona, F. Petit and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, in Exponential functionals and principal values related to Brownian motion (ed. Marc Yor), Rev. Mat. Iberoamericana, Madrid, (1997), 73-130. Google Scholar

[14]

M. Doumic and M. Escobedo, Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297. doi: 10.3934/krm.2016.9.251. Google Scholar

[15]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001. Google Scholar

[16]

A. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6 (1961), 275-294. doi: 10.1137/1106036. Google Scholar

[17]

C. M. Goldie and R. A. Maller, Stability of perpetuities, Ann. Probab., 28 (2000), 1195-1218. doi: 10.1214/aop/1019160331. Google Scholar

[18]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Process. Appl., 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0. Google Scholar

[19]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429. doi: 10.1214/09-AAP622. Google Scholar

[20]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[21]

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

[22]

D. J. McLaughlinW. Lamb and A. C. McBride, A semigroup approach to fragmentation models, SIAM J. Math. Anal., 28 (1997), 1158-1172. doi: 10.1137/S0036141095291701. Google Scholar

[23]

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

[24]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001. Google Scholar

[25]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898. doi: 10.1016/j.anihpc.2015.01.007. Google Scholar

[26]

T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation, Aerosol Sci. Technol., 5 (1986), 93-101. doi: 10.1080/02786828608959079. Google Scholar

[27]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math., 43 (1995), 245-262. Google Scholar

[28]

M. Tyran-Kamińska, Substochastic semigroups and densities of piecewise deterministic Markov processes, J. Math. Anal. Appl., 357 (2009), 385-402. doi: 10.1016/j.jmaa.2009.04.033. Google Scholar

[29]

W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab., 11 (1979), 750-783. doi: 10.2307/1426858. Google Scholar

[30]

W. Wagner, Random and deterministic fragmentation models, Monte Carlo Methods Appl., 16 (2010), 399-420. doi: 10.1515/MCMA.2010.016. Google Scholar

[31]

R. Wieczorek, A stochastic particles model of fragmentation process with shattering, Electron. J. Probab., 20 (2015), 17pp. doi: 10.1214/EJP.v20-4060. Google Scholar

[32]

R. M. Ziff, New solutions to the fragmentation equation, J. Phys. A, 24 (1991), 2821-2828. doi: 10.1088/0305-4470/24/12/020. Google Scholar

[33]

R. M. Ziff and E. McGrady, Kinetics of polymer degradation, Macromolecules, 19 (1986), 2513-2519. doi: 10.1021/ma00164a010. Google Scholar

show all references

References:
[1]

O. Arino and R. Rudnicki, Stability of phytoplankton dynamics, C. R. Biologies, 327 (2004), 961-969. doi: 10.1016/j.crvi.2004.03.013. Google Scholar

[2]

L. Arlotti and J. Banasiak, Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss, J. Math. Anal. Appl., 293 (2004), 693-720. doi: 10.1016/j.jmaa.2004.01.028. Google Scholar

[3]

J. Banasiak, On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math., 5 (2001), 169-191. doi: 10.11650/twjm/1500574893. Google Scholar

[4]

J. Banasiak, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci., 14 (2004), 483-501. doi: 10.1142/S0218202504003325. Google Scholar

[5]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications Springer-Verlag London Ltd., London, 2006. doi: 10.1007/1-84628-153-9. Google Scholar

[6]

J. Banasiak and W. Lamb, On the application of substochastic semigroup theory to fragmentation models with mass loss, J. Math. Anal. Appl., 284 (2003), 9-30. doi: 10.1016/S0022-247X(03)00154-9. Google Scholar

[7]

J. Banasiak and M. Mokhtar-Kharroubi, Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 529-542. doi: 10.3934/dcdsb.2005.5.529. Google Scholar

[8]

J. Bertoin, Random Fragmentation and Coagulation Processes Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511617768. Google Scholar

[9]

J. Bertoin and M.-E. Caballero, Entrance from 0+ for increasing semi-stable Markov processes, Bernoulli, 8 (2002), 195-205. Google Scholar

[10]

J. Bertoin and A. R. Watson, Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61. doi: 10.1017/apr.2016.41. Google Scholar

[11]

W. Biedrzycka and M. Tyran-Kamińska, Existence of invariant densities for semiflows with jumps, J. Math. Anal. Appl., 435 (2016), 61-84. doi: 10.1016/j.jmaa.2015.10.019. Google Scholar

[12]

M. E. Caballero and V. Rivero, On the asymptotic behaviour of increasing self-similar Markov processes, Electron. J. Probab., 14 (2009), 865-894. doi: 10.1214/EJP.v14-637. Google Scholar

[13]

P. Carmona, F. Petit and M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, in Exponential functionals and principal values related to Brownian motion (ed. Marc Yor), Rev. Mat. Iberoamericana, Madrid, (1997), 73-130. Google Scholar

[14]

M. Doumic and M. Escobedo, Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297. doi: 10.3934/krm.2016.9.251. Google Scholar

[15]

M. EscobedoS. Mischler and M. Rodriguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001. Google Scholar

[16]

A. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6 (1961), 275-294. doi: 10.1137/1106036. Google Scholar

[17]

C. M. Goldie and R. A. Maller, Stability of perpetuities, Ann. Probab., 28 (2000), 1195-1218. doi: 10.1214/aop/1019160331. Google Scholar

[18]

B. Haas, Loss of mass in deterministic and random fragmentations, Stochastic Process. Appl., 106 (2003), 245-277. doi: 10.1016/S0304-4149(03)00045-0. Google Scholar

[19]

B. Haas, Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429. doi: 10.1214/09-AAP622. Google Scholar

[20]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. Google Scholar

[21]

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

[22]

D. J. McLaughlinW. Lamb and A. C. McBride, A semigroup approach to fragmentation models, SIAM J. Math. Anal., 28 (1997), 1158-1172. doi: 10.1137/S0036141095291701. Google Scholar

[23]

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

[24]

P. MichelS. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001. Google Scholar

[25]

S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898. doi: 10.1016/j.anihpc.2015.01.007. Google Scholar

[26]

T. W. Peterson, Similarity solutions for the population balance equation describing particle fragmentation, Aerosol Sci. Technol., 5 (1986), 93-101. doi: 10.1080/02786828608959079. Google Scholar

[27]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math., 43 (1995), 245-262. Google Scholar

[28]

M. Tyran-Kamińska, Substochastic semigroups and densities of piecewise deterministic Markov processes, J. Math. Anal. Appl., 357 (2009), 385-402. doi: 10.1016/j.jmaa.2009.04.033. Google Scholar

[29]

W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab., 11 (1979), 750-783. doi: 10.2307/1426858. Google Scholar

[30]

W. Wagner, Random and deterministic fragmentation models, Monte Carlo Methods Appl., 16 (2010), 399-420. doi: 10.1515/MCMA.2010.016. Google Scholar

[31]

R. Wieczorek, A stochastic particles model of fragmentation process with shattering, Electron. J. Probab., 20 (2015), 17pp. doi: 10.1214/EJP.v20-4060. Google Scholar

[32]

R. M. Ziff, New solutions to the fragmentation equation, J. Phys. A, 24 (1991), 2821-2828. doi: 10.1088/0305-4470/24/12/020. Google Scholar

[33]

R. M. Ziff and E. McGrady, Kinetics of polymer degradation, Macromolecules, 19 (1986), 2513-2519. doi: 10.1021/ma00164a010. Google Scholar

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