Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | 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 [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.2307/3318611
  • [2] 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.
  • [3] 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.
  • [4] 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, https://doi.org/10.1016/j.anihpc.2004.06.001
  • [5] 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, https://doi.org/10.1007/s00220-004-1258-5
  • [6] S.K. Friedlander, Smoke, dust and haze: Fundamentals of aerosol dynamics, second ed., Topics in Chemical Engineering, Oxford University Press, 2000.
  • [7] 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, https://doi.org/10.1090/qam/1454
  • [8] P. Laurençot, B. Niethammer, and J.J.L. Velázquez, Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel, (2016), Preprint, arxiv:1603:02929.
  • [9] 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, https://doi.org/10.1007/s10955-011-0239-2
  • [10] 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, https://doi.org/10.1002/cpa.3048
  • [11] 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, https://doi.org/10.1007/s10955-016-1553-5
  • [12] M Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, Physik. Zeitschrift 17 (1916), 557-599.
  • [13] 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, https://doi.org/10.1007/BF01022996

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 70F99, 82C22, 45M10

Retrieve articles in all journals with MSC (2010): 70F99, 82C22, 45M10


Additional Information

B. Niethammer
Affiliation: Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: niethammer@iam.uni-bonn.de

J. J. L. Velázquez
Affiliation: Institute of Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: velazquez@iam.uni-bonn.de

DOI: https://doi.org/10.1090/qam/1478
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

American Mathematical Society