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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 5(389), Pages 97–166 (Mi umn9308)  

This article is cited in 6 scientific papers (total in 6 papers)

Scattering in billiards and problems of Newtonian aerodynamics

A. Yu. Plakhovab

a University of Aveiro
b Aberystwyth University

Abstract: This paper contains results relating to billiards and their applications to various resistance optimization problems generalizing Newton's aerodynamic problem. The results can be divided into three groups. First, minimum resistance problems for bodies moving translationally in a highly rarefied medium are considered. It is shown that generically the infimum of the resistance is zero, that is, there are almost ‘perfectly streamlined’ bodies. Second, a rough body is defined and results on characterization of billiard scattering on non-convex and rough bodies are presented. Third, these results are used to reduce some problems on minimum and maximum resistance of moving and slowly rotating bodies to special problems on optimal mass transfer, which are then explicitly solved. In particular, the resistance of a 3-dimensional convex body can be at most doubled or at most reduced by 3.05% by grooving its surface.
Bibliography: 27 titles.

Keywords: billiards, scattering, Newton's aerodynamic problem, optimal mass transfer, free molecular flow, rough body.

DOI: https://doi.org/10.4213/rm9308

Full text: PDF file (1220 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2009, 64:5, 873–938

Bibliographic databases:

UDC: 517.987+517.972
MSC: Primary 49Q10, 76G25, 37D50; Secondary 49Q20, 76M28
Received: 07.06.2009

Citation: A. Yu. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics”, Uspekhi Mat. Nauk, 64:5(389) (2009), 97–166; Russian Math. Surveys, 64:5 (2009), 873–938

Citation in format AMSBIB
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\by A.~Yu.~Plakhov
\paper Scattering in billiards and problems of Newtonian aerodynamics
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 5(389)
\pages 97--166
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\transl
\jour Russian Math. Surveys
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\vol 64
\issue 5
\pages 873--938
\crossref{https://doi.org/10.1070/RM2009v064n05ABEH004642}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Plakhov A., Roshchina V., “Invisibility in billiards”, Nonlinearity, 24:3 (2011), 847–854  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Plakhov A., “Mathematical retroreflectors”, Discrete Contin. Dyn. Syst., 30:4 (2011), 1211–1235  crossref  mathscinet  zmath  isi  scopus
    3. Plakhov A., “Billiards, scattering by rough obstacles, and optimal mass transportation”, J. Math. Sci., 182:2 (2012), 233–245  crossref  mathscinet  zmath  scopus
    4. Roshchina V., Plakhov F., “Fractal bodies invisible in 2 and 3 directions”, Discrete and Continuous Dynamical Systems Ser. A, 33:4 (2012), 1615–1631  crossref  isi  scopus
    5. Plakhov A., “Optimal roughening of convex bodies”, Canad. J. Math., 64:5 (2012), 1058–1074  crossref  mathscinet  zmath  isi  scopus
    6. Galperin G., Plakhov A., “Billiards, Invisibility, and Perfectly Streamlining Objects”, Interdiscip. Stud. Complex Syst., 2017, no. 9, 25–35  isi
  • Успехи математических наук Russian Mathematical Surveys
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