Existence and uniqueness of solutions of Smoluchowski's coagulation equation with source terms

Authors:
M. Shirvani and H. J. Van Roessel

Journal:
Quart. Appl. Math. **60** (2002), 183-194

MSC:
Primary 82D60; Secondary 34C60, 82C05

DOI:
https://doi.org/10.1090/qam/1878265

MathSciNet review:
MR1878265

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence and uniqueness of a solution to the Smoluchowski coagulation equation with source terms. The coagulation equation with source terms is potentially useful in applications because one sometimes tries to control coagulation processes by the introduction of particles of various sizes into the system. The existence proof given here differs in style from most other existence proofs in two respects. First, it is not based on a finite-dimensional truncation of the coagulation equation; and secondly, it is achieved with a weaker hypothesis than is usually assumed on the initial data.

**[1]**Milton Abramowitz and Irene A. Stegun,*Handbook of Mathematical Functions*, Dover, 1965**[2]**J. M. Ball and J. Carr,*Asymptotic behaviour of solutions to the Becker-Döring cluster equations for arbitrary initial data*, Proc. Roy. Soc. Edinburgh**108A**, 109-116 (1988) MR**931012****[3]**J. M. Ball and J. Carr,*The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation*, J. Stat. Phys.**61**, 203-234 (1990) MR**1084278****[4]**J. M. Ball, J. Carr, and O. Penrose,*The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions*, Comm. Math. Phys.**104**, 657-692 (1986) MR**841675****[5]**J. D. Barrow,*Coagulation with fragmentation*, J. Phys. A**14**, 729-733 (1981) MR**605266****[6]**R. J. Cohen and G. B. Benderek,*Equilibrium and kinetic theory of sol-gel transition*, J. Chem. Phys.**86**, 3696-3714 (1982)**[7]**F. P. da Costa,*Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentation*, J. Math. Anal. Appl.**192**, 892-914 (1995) MR**1336484****[8]**R. L. Drake,*A general mathematical survey of the coagulation equation*, Topics in Current Aerosol Research 3 (Part 2) (G. Hidy and J. R. Brock, eds.), Pergamon Press, 1972**[9]**P. B. Dubovskii,*Convergence of the solutions of the coagulation equation with a source to an equilibrium state*, Differential Equations**31**, 635-640 (1995) MR**1370336****[10]**Jack K. Hale,*Ordinary Differential Equations*, second edition, Krieger Publishing Co., Huntington, NY, 1980 MR**587488****[11]**E. M. Hendriks, M. H. Ernst, and R. M. Ziff,*Coagulation equations with gelation*, J. Stat. Phys.**31**, 519-563 (1983) MR**711489****[12]**F. Leyvraz,*Existence and properties of post-gel solutions for the kinetic equations of coagulation*, J. Phys. A: Math. Gen.**16**, 2861-2873 (1983) MR**715741****[13]**F. Leyvraz and H. R. Tschudi,*Singularities in kinetics of coagulation processes*, J. Phys. A: Math. Gen.**14**, 3389-3405 (1981) MR**639565****[14]**J. B. McLeod,*On a recurrence formula in differential equations*, Quart. J. Math. Oxford Ser. (2)**13**, 283-284 (1962) MR**0143976****[15]**J. B. McLeod,*On an infinite set of non-linear differential equations*, Quart J. Math. Oxford Ser. (2)**13**, 119-128 (1962) MR**0139822****[16]**J. B. McLeod,*On an infinite set of non-linear differential equations*(II), Quart. J. Math. Oxford Ser. (2)**13**, 193-205 (1962) MR**0140745****[17]**R. W. Samsel and A. S. Perelson,*Kinetics of rouleaux formation*, Biophys. J.**37**, 493-514 (1982)**[18]**M. Shirvani and H. J. Van Roessel,*The mass-conserving solutions of Smoluchowski's coagulation equation: The general bilinear kernel*, Z. Angew. Math. Phys.**43**, 526-535 (1992) MR**1166971****[19]**M. Slemrod,*Trend to equilibrium in the Becker-Döring cluster equations*, Nonlinearity**2**, 429-443 (1989) MR**1005058****[20]**M. von Smoluchowski,*Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen*, Z. Phys.**17**, 557-585 (1916)**[21]**M. von Smoluchowski,*Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen*, Z. Phys. Chem.**92**, 124-168 (1917)**[22]**John L. Spouge,*An existence theorem for the discrete coagulation-fragmentation equations*, Math. Proc. Cambridge Philos. Soc.**96**, 351-357 (1984) MR**757669****[23]**W. H. White,*A global existence theorem for Smoluchowski's coagulation equations*, Proc. Amer. Math. Soc.**80**, 273-276 (1980) MR**577758**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
82D60,
34C60,
82C05

Retrieve articles in all journals with MSC: 82D60, 34C60, 82C05

Additional Information

DOI:
https://doi.org/10.1090/qam/1878265

Article copyright:
© Copyright 2002
American Mathematical Society