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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[33]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[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.

[23]

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

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[33]

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

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