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Local well posedness for a linear coagulation equation
Authors:
M. Escobedo and J. J. L. Velázquez
Journal:
Trans. Amer. Math. Soc. 365 (2013), 1743-1808
MSC (2010):
Primary 45K05, 45A05, 45M05, 82C40, 82C05, 82C22
Posted:
October 4, 2012
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Abstract: In this paper a family of linear coagulation models is solved. These models arise in the analysis of the asymptotic behaviour of coagulation equations yielding gelation for large particles. The tools and techniques that are developed in this paper are based on the definition of a class of weighted Sobolev spaces that take into account the characteristic time scales associated to the coagulation equation for large particles, as well as in the continuation argument introduced by Schauder to prove well posedness of general classes of elliptic and parabolic equations.
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- A. M. Balk & V. E. Zakharov, Stability of Weak-Turbulence Kolmogorov Spectra, in Nonlinear Waves and Weak Turbulence, V. E. Zakharov ed., A. M. S. Translations Series 2, 182, 1-81 (1998). MR 1618499 (99e:76057)
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- L. Desvillettes, About the regularizing properties of the non-cut-off Kac equation, Comm. Math. Phys. 168, 417-440 (1995). MR 1324404 (96d:82052)
- 3.
- M. H. Ernst, R. M. Ziff & E. M. Hendriks, Coagulation processes with a phase transition, J. of Colloid and Interface Sci. 97, 266-277 (1984).
- 4.
- M. Escobedo & J. J. L. Velázquez, On the Fundamental Solution of a Homogeneous Linearized Coagulation Equation. Comm. Math. Phys. 297, 759-816 (2010). MR 2653902
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- M. Escobedo & J. J. L. Velázquez, Classical non mass preserving solutions of coagulation equations. Preprint.
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- D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order. 2nd Edition. Springer 1983. MR 737190 (86c:35035)
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- O. A. Ladyzhenskaja and N. N. Ural'tseva, Linear and quasilinear ellitic equations, Academic Press 1968. MR 0244627 (39:5941)
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- F. Leyvraz, Scaling Theory and Exactly Solved Models in the Kintetics of Irreversible Aggregation, Phys. Reports 383, Issues 2-3, 95-212 (2003).
- 9.
- J. B. McLeod, On the scalar transport equation Proc. London Math. Soc. 14, 445-458 (1964). MR 0162110 (28:5311)
- 10.
- G. Menon & R. L. Pego, Approach to self-similarity in Smoluchowski's coagulation equations, Comm. Pure. Appl. Math. 57, 1197-1232 (2004). MR 2059679 (2005i:82051)
- 11.
- G. Menon & R. L. Pego, Dynamical scaling in Smoluchowski's coagulation equation: Uniform convergence, SIAM J. Math. Anal. 36, 1629-1651 (2005). MR 2139564 (2006e:35330)
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- H. Tanaka, S. Inaba & K. Nakazawa Steady-State Size Distribution for the Self-Similar Collision Cascade ICARUS 123, 450-455 (1996).
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- P. G. J. van Dongen & M.H. Ernst, Cluster size distribution in irreversible aggregation at large times, J. Phys. A 18, 2779-2793 (1985) . MR 811992 (87f:82004)
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- C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics (Vol. 1), S. Friedlander and D. Serre ed., Elsevier Science (2002). MR 1942465 (2003k:82087)
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- W. von Wahl, The continuity or stability method for nonlinear eliptic and parabolic equations and systems, Milan J. of Math. 62, 157-183 (1992). MR 1293779 (95i:35153)
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Additional Information
M. Escobedo
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, E–48080 Bilbao, Spain
Email:
mtpesmam@lg.ehu.es
J. J. L. Velázquez
Affiliation:
ICMAT (CSIC-UAM-UC3M-UCM) Facultad de Matemáticas, Universidad Complutense, E–28040 Madrid, Spain
Address at time of publication:
Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email:
JJ{\textunderscore}Velazquez@mat.ucm.es, velazquez@iam.uni{\textunderscore}bonn.de
DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05576-0
PII:
S 0002-9947(2012)05576-0
Received by editor(s):
August 3, 2010
Received by editor(s) in revised form:
February 23, 2011
Posted:
October 4, 2012
Article copyright:
© Copyright 2012 American Mathematical Society
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