Abstract
Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher-order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when fragmentation is described by power-law rates, which are commonly used in engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L 1 spaces.
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Johansen A, Brauer C, Dullemond CP, Klahr H, Henning T (2008) A coagulation–fragmentation model for the turbulent growth and destruction of preplanetesimals. Astron Astrophys 486: 597–613
Samsel RW, Perelson AS (1982) Kinetics of roleau formation. Biophys J 37: 493–514
Vigil RD, Ziff RM (1989) On the stability of coagulation–fragmentation population balances. J Colloid Interface Sci 133: 257–264
Ziff RM (1980) Kinetics of polymerization. J Stat Phys 23: 241–263
Amar JG, Popescu MN, Family F (2001) Rate-equation approach to island capture zones and size distributions in epitaxial growth. Phys Rev Lett 86: 3092–3095
McGrady ED, Ziff RM (1987) “Shattering” transition in fragmentation. Phys Rev Lett 58: 892–895
McLaughlin DJ, Lamb W, McBride AC (1997) A semigroup approach to fragmentation models. SIAM J Math Anal 28: 1158–1172
McLaughlin DJ, Lamb W, McBride AC (1997) An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J Math Anal 28: 1173–1190
Banasiak J, Arlotti L (2006) Perturbations of positive semigroups with applications. Springer, London
Banasiak J, Oukouomi Noutchie SC, Rudnicki R (2009) Global solvability of a fragmentation–coagulation equation with growth and restricted coagulation. J Nonlinear Math Phys 16(1): 13–26
Stewart IW (1989) A global existence theorem for the global coagulation–fragmentation equation with unbounded kernels. Math Methods Appl Sci 11: 627–648
Dubovskiĭ PB, Stewart IW (1996) Existence, uniqueness and mass conservation for the coagulation–fragmentation equation. Math Methods Appl Sci 19: 571–591
Laurençot P (2000) On a class of continuous coagulation–fragmentation equations. J Differ Equ 167: 245–274
Giri AK, Kumar J, Warnecke G (2011) The continuous coagulation equation with multiple fragmentation. J Math Anal Appl 374: 71–87
Banasiak J, Lamb W (2011) Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation. Proc R Soc Edinb A 141: 465–480
Aizenman M, Bak TA (1979) Convergence to equilibrium in a system of reacting polymers. Commun Math Phys 65: 203–230
Banasiak J, Lamb W (2009) Coagulation, fragmentation and growth processes in a size structured population. Discrete Contin Dyn Syst B 11: 563–585
Banasiak J (2012) Transport processes with coagulation and strong fragmentation. Discrete Contin Dyn Syst B 17: 445–472
Lamb W (2004) Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math Methods Appl Sci 27: 703–721
Lunardi A (1995) Analytic semigroups and optimal regularity in parabolic problems. Birkhäuser-Verlag, Basel
Aldous DJ (1999) Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5: 3–48
Smit DJ, Hounslow MJ, Paterson WR (1994) Aggregation and gelation—I. Analytical solutions for CST and batch operation. Chem Eng Sci 49: 1025–1035
Koch J, Hackbusch W, Sundmacher K (2007) H-matrix methods for linear and quasi-linear integral operators appearing in population balances. Comput Chem Eng 31: 745–759
Banasiak J (2012) Global classical solutions of coagulation–fragmentation equations with unbounded coagulation rates. Nonlinear Anal Real World Appl 13: 91–105
Banasiak J, Lamb W (2012) Analytic fragmentation semigroups and continuous coagulation–fragmentation equations with unbounded rates. J Math Anal Appl 391: 312–322
Banasiak J, Oukoumi Noutchie SC (2010) Controlling number of particles in fragmentation equations. Physica D 239: 1422–1435
Edwards BF, Cai M, Han H (1990) Rate equation and scaling for fragmentation with mass loss. Phys Rev A 41: 5755–5757
Smith L, Lamb W, Langer M, McBride A (2012) Discrete fragmentation with mass loss. J Evol Equ 12: 181–201
Hardy GH, Littlewood JE, Pólya G (1934) Inequalities. Cambridge University Press, Cambridge
Triebel H (1978) Interpolation theory, function spaces, differential operators. North Holland, Amsterdam
Banasiak J (2002) On a non-uniqueness in fragmentation models. Math Methods Appl Sci 25: 541–556
Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, New York
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This paper is dedicated to Professor P. G. L. Leach on his 70th birthday.
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Banasiak, J., Lamb, W. & Langer, M. Strong fragmentation and coagulation with power-law rates. J Eng Math 82, 199–215 (2013). https://doi.org/10.1007/s10665-012-9596-3
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DOI: https://doi.org/10.1007/s10665-012-9596-3