Oscillatory traveling wave solutions for coagulation equations
Authors:
B. Niethammer and J. J. L. Velázquez
Journal:
Quart. Appl. Math. 76 (2018), 153-188
MSC (2010):
Primary 70F99, 82C22, 45M10
DOI:
https://doi.org/10.1090/qam/1478
Published electronically:
August 1, 2017
MathSciNet review:
3733098
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Additional Information
Abstract: We consider Smoluchowski’s coagulation equation with kernels of homogeneity one of the form $K_{\varepsilon }(\xi ,\eta ) =\big ( \xi ^{1-\varepsilon }+\eta ^{1-\varepsilon }\big )\big ( \xi \eta \big ) ^{\frac {\varepsilon }{2}}$. Heuristically, in suitable exponential variables, one can argue that in this case the long-time behaviour of solutions is similar to the inviscid Burgers equation and that for Riemann data solutions converge to a traveling wave for large times. Numerical simulations in a work by Herrmann and the authors indeed support this conjecture, but also reveal that the traveling waves are oscillatory and the oscillations become stronger with smaller $\varepsilon$. The goal of this paper is to construct such oscillatory traveling wave solutions and provide details of their shape via formal matched asymptotic expansions.
References
- David J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli 5 (1999), no. 1, 3–48. MR 1673235, DOI https://doi.org/10.2307/3318611
- M. Bonacini, B. Niethammer, and Velázquez J.J.L., Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one, (2016), Preprint, arxiv:1612.06610.
- R.-L. Drake, A general mathematical survey of the coagulation equation, Topics in current aerosol research (part 2), Hidy G. M., Brock, J. R. eds., International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, 1972, pp. 203–376.
- 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), no. 1, 99–125 (English, with English and French summaries). MR 2114413, DOI https://doi.org/10.1016/j.anihpc.2004.06.001
- Nicolas Fournier and Philippe Laurençot, Existence of self-similar solutions to Smoluchowski’s coagulation equation, Comm. Math. Phys. 256 (2005), no. 3, 589–609. MR 2161272, DOI https://doi.org/10.1007/s00220-004-1258-5
- S.K. Friedlander, Smoke, dust and haze: Fundamentals of aerosol dynamics, second ed., Topics in Chemical Engineering, Oxford University Press, 2000.
- Michael Herrmann, Barbara Niethammer, and Juan J. L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math. 75 (2017), no. 1, 105–130. MR 3580097, DOI https://doi.org/10.1090/qam/1454
- P. Laurençot, B. Niethammer, and J.J.L. Velázquez, Oscillatory dynamics in Smoluchowski’s coagulation equation with diagonal kernel, (2016), Preprint, arxiv:1603:02929.
- J. B. McLeod, B. Niethammer, and J. J. L. Velázquez, Asymptotics of self-similar solutions to coagulation equations with product kernel, J. Stat. Phys. 144 (2011), no. 1, 76–100. MR 2820036, DOI https://doi.org/10.1007/s10955-011-0239-2
- Govind Menon and Robert L. Pego, Approach to self-similarity in Smoluchowski’s coagulation equations, Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232. MR 2059679, DOI https://doi.org/10.1002/cpa.3048
- B. Niethammer, S. Throm, and J. J. L. Velázquez, A uniqueness result for self-similar profiles to Smoluchowski’s coagulation equation revisited, J. Stat. Phys. 164 (2016), no. 2, 399–409. MR 3513258, DOI https://doi.org/10.1007/s10955-016-1553-5
- M Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschrift 17 (1916), 557–599.
- P. G. J. van Dongen and M. H. Ernst, Scaling solutions of Smoluchowski’s coagulation equation, J. Statist. Phys. 50 (1988), no. 1-2, 295–329. MR 939490, DOI https://doi.org/10.1007/BF01022996
References
- David J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists, Bernoulli 5 (1999), no. 1, 3–48. MR 1673235, DOI https://doi.org/10.2307/3318611
- M. Bonacini, B. Niethammer, and Velázquez J.J.L., Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity one, (2016), Preprint, arxiv:1612.06610.
- R.-L. Drake, A general mathematical survey of the coagulation equation, Topics in current aerosol research (part 2), Hidy G. M., Brock, J. R. eds., International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, 1972, pp. 203–376.
- 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), no. 1, 99–125 (English, with English and French summaries). MR 2114413, DOI https://doi.org/10.1016/j.anihpc.2004.06.001
- Nicolas Fournier and Philippe Laurençot, Existence of self-similar solutions to Smoluchowski’s coagulation equation, Comm. Math. Phys. 256 (2005), no. 3, 589–609. MR 2161272, DOI https://doi.org/10.1007/s00220-004-1258-5
- S.K. Friedlander, Smoke, dust and haze: Fundamentals of aerosol dynamics, second ed., Topics in Chemical Engineering, Oxford University Press, 2000.
- Michael Herrmann, Barbara Niethammer, and Juan J. L. Velázquez, Instabilities and oscillations in coagulation equations with kernels of homogeneity one, Quart. Appl. Math. 75 (2017), no. 1, 105–130. MR 3580097, DOI https://doi.org/10.1090/qam/1454
- P. Laurençot, B. Niethammer, and J.J.L. Velázquez, Oscillatory dynamics in Smoluchowski’s coagulation equation with diagonal kernel, (2016), Preprint, arxiv:1603:02929.
- J. B. McLeod, B. Niethammer, and J. J. L. Velázquez, Asymptotics of self-similar solutions to coagulation equations with product kernel, J. Stat. Phys. 144 (2011), no. 1, 76–100. MR 2820036, DOI https://doi.org/10.1007/s10955-011-0239-2
- Govind Menon and Robert L. Pego, Approach to self-similarity in Smoluchowski’s coagulation equations, Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232. MR 2059679, DOI https://doi.org/10.1002/cpa.3048
- B. Niethammer, S. Throm, and J. J. L. Velázquez, A uniqueness result for self-similar profiles to Smoluchowski’s coagulation equation revisited, J. Stat. Phys. 164 (2016), no. 2, 399–409. MR 3513258, DOI https://doi.org/10.1007/s10955-016-1553-5
- M Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschrift 17 (1916), 557–599.
- P. G. J. van Dongen and M. H. Ernst, Scaling solutions of Smoluchowski’s coagulation equation, J. Statist. Phys. 50 (1988), no. 1-2, 295–329. MR 939490, DOI https://doi.org/10.1007/BF01022996
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Additional Information
B. Niethammer
Affiliation:
Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
MR Author ID:
359693
Email:
niethammer@iam.uni-bonn.de
J. J. L. Velázquez
Affiliation:
Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
MR Author ID:
289301
Email:
velazquez@iam.uni-bonn.de
Keywords:
Smoluchowski’s coagulation equation,
kernels with homogeneity one,
traveling waves
Received by editor(s):
February 8, 2017
Received by editor(s) in revised form:
June 16, 2017
Published electronically:
August 1, 2017
Article copyright:
© Copyright 2017
Brown University
