×

Existence and nonexistence for the collision-induced breakage equation. (English) Zbl 1473.45014

Summary: A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter distribution functions, assuming the collision kernel \(K\) to be given by \(K(x,y)= x^{\alpha} y^{\beta} + x^{\beta} y^{\alpha}\) with \(\alpha \leq \beta \leq 1\). When \(\alpha + \beta \in [1,2]\), it is shown that there exists at least one weak mass-conserving solution for all times. In contrast, when \(\alpha + \beta \in [0,1)\) and \(\alpha \geq 0\), global mass-conserving weak solutions do not exist, though such solutions are constructed on a finite time interval depending on the initial condition. The question of uniqueness is also considered. Finally, for \(\alpha <0\) and a specific daughter distribution function, the nonexistence of mass-conserving solutions is also established.

MSC:

45K05 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. Amann, Ordinary Differential Equations: An Introduction to Nonlinear Analysis, De Gruyter Stud. Math. 13, Walter de Gruyter, Berlin, 1990, https://doi.org/10.1515/9783110853698. · Zbl 0708.34002
[2] J. Banasiak, W. Lamb, and Ph. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, CRC Press, Boca Raton, FL, 2019. · Zbl 1434.82001
[3] J. Bertoin, Random Fragmentation and Coagulation Processes, Cambridge Stud. Adv. Math. 102, Cambridge University Press, Cambridge, UK, 2006, https://doi.org/10.1017/CBO9780511617768. · Zbl 1107.60002
[4] N. Brilliantov, P. L. Krapivsky, A. Bodrova, F. Spahn, H. Hayakawa, V. Stadnichuk, and J. Schmidt, Size distribution of particles in Saturn\textquoterights rings from aggregation and fragmentation, Proc. Natl. Acad. Sci. USA, 112 (2015), pp. 9536-9541, https://doi.org/10.1073/pnas.1503957112.
[5] J. Carr and F. P. da Costa, Instantaneous gelation in coagulation dynamics, Z. Angew. Math. Phys., 43 (1992), pp. 974-983, https://doi.org/10.1007/BF00916423. · Zbl 0761.76011
[6] Z. Cheng and S. Redner, Scaling theory of fragmentation, Phys. Rev. Lett., 60 (1988), pp. 2450-2453.
[7] Z. Cheng and S. Redner, Kinetics of fragmentation, J. Phys. A, 23 (1990), pp. 1233-1258, http://stacks.iop.org/0305-4470/23/1233.
[8] C. De La Vallée Poussin, Sur l’intégrale de Lebesgue, Trans. Amer. Math. Soc., 16 (1915), pp. 435-501, https://doi.org/10.2307/1988879. · JFM 45.0441.06
[9] M. Ernst and I. Pagonabarraga, The nonlinear fragmentation equation, J. Phys. A, 40 (2007), pp. F331-F337, https://doi.org/10.1088/1751-8113/40/17/F03. · Zbl 1189.82068
[10] M. Escobedo, S. 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), pp. 99-125, https://doi.org/10.1016/j.anihpc.2004.06.001. · Zbl 1130.35025
[11] A. F. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6 (1961), pp. 275-294. · Zbl 0242.60050
[12] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: \(L^p\) Spaces, Springer Monogr. Math., Springer, New York, 2007. · Zbl 1153.49001
[13] A. K. Giri, On the uniqueness for coagulation and multiple fragmentation equation, Kinet. Relat. Models, 6 (2013), pp. 589-599, https://doi.org/10.3934/krm.2013.6.589. · Zbl 1264.45014
[14] A. K. Giri and Ph. Laurençot, Weak solutions to the collision-induced breakage equation with dominating coagulation, J. Differential Equations, 280 (2021), pp. 690-729, https://doi.org/10.1016/j.jde.2021.01.043. · Zbl 1464.45019
[15] P. Kapur, Self-preserving size spectra of comminuted particles, Chem. Eng. Sci., 27 (1972), pp. 425-431, https://doi.org/https://doi.org/10.1016/0009-2509(72)85079-6.
[16] M. Kostoglou and A. J. Karabelas, A study of the nonlinear breakage equation: Analytical and asymptotic solutions, J. Phys. A, 33 (2000), pp. 1221-1232, https://doi.org/10.1088/0305-4470/33/6/309. · Zbl 0969.82026
[17] M. Kostoglou and A. J. Karabelas, A study of the collisional fragmentation problem using the gamma distribution approximation, J. Colloid Interface Sci., 303 (2006), pp. 419-429.
[18] P. L. Krapivsky and E. Ben-Naim, Shattering transitions in collision-induced fragmentation, Phys. Rev. E, 68 (2003), 021102, https://doi.org/10.1103/PhysRevE.68.021102.
[19] R. List and J. R. Gillespie, Evolution of raindrop spectra with collision-induced breakup, J. Atmos. Sci., 33 (1976), pp. 2007-2013, https://doi.org/10.1175/1520-0469(1976)033<2007:EORSWC>2.0.CO;2.
[20] E. D. McGrady and R. M. Ziff, “Shattering” transition in fragmentation, Phys. Rev. Lett., 58 (1987), pp. 892-895, https://doi.org/10.1103/PhysRevLett.58.892.
[21] V. Safronov, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets, Israel Program for Scientific Translations, Jerusalem, 1972.
[22] R. C. Srivastava, Parameterization of raindrop size distributions, J. Atmos. Sci., 35 (1978), pp. 108-117, https://doi.org/10.1175/1520-0469(1978)035<0108:PORSD>2.0.CO;2.
[23] I. W. Stewart, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels, Math. Methods Appl. Sci., 11 (1989), pp. 627-648, https://doi.org/10.1002/mma.1670110505. · Zbl 0683.45006
[24] I. W. Stewart, A uniqueness theorem for the coagulation-fragmentation equation, Math. Proc. Cambridge Philos. Soc., 107 (1990), pp. 573-578, https://doi.org/10.1017/S0305004100068821. · Zbl 0708.45010
[25] P. G. J. van Dongen, On the possible occurrence of instantaneous gelation in Smoluchowski’s coagulation equation, J. Phys. A, 20 (1987), pp. 1889-1904.
[26] R. Vigil, I. Vermeersch, and R. Fox, Destructive aggregation: Aggregation with collision-induced breakage, J. Colloid Interface Sci., 302 (2006), pp. 149-158, https://doi.org/10.1016/j.jcis.2006.05.066.
[27] I. I. Vrabie, \(C_0\)-Semigroups and Applications, North-Holland Math. Stud. 191, North-Holland, Amsterdam, 2003. · Zbl 1119.47044
[28] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerisation, J. Phys. A, 18 (1985), pp. 3027-3037, http://stacks.iop.org/0305-4470/18/3027.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.