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
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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.

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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

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