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Mat. Sb., 2004, Volume 195, Number 7, Pages 105–126 (Mi msb836)  

This article is cited in 14 scientific papers (total in 14 papers)

Newton's problem of the body of minimum mean resistance

A. Yu. Plakhov

University of Aveiro

Abstract: Consider a body $\Omega$ at rest in $d$-dimensional Euclidean space and a homogeneous flow of particles falling on it with unit velocity $v$. The particles do not interact and they collide with the body perfectly elastically. Let $\mathscr R_\Omega(v)$ be the resistance of the body to the flow. The problem of the body of minimum resistance, which goes back to Newton, consists in the minimization of the quantity $(\mathscr R_\Omega(v)\mid v)$ over a prescribed class of bodies.
Assume that one does not know in advance the direction $v$ of the flow or that one measures the resistance repeatedly for various directions of $v$. Of interest in these cases is the problem of the minimization of the mean value of the resistance $\widetilde{\mathscr R}(\Omega) =\displaystyle\int_{S^{d-1}}(\mathscr R_\Omega(v)\mid v) dv$. This problem is considered $(\widetilde{\mathrm{P}}_d)$ in the class of bodies of volume 1 and $(\widetilde{\mathrm{P}} _d^c)$ in the class of convex bodies of volume 1. The solution of the convex problem $\widetilde{\mathrm{P}} _d^c$ is the $d$-dimensional ball. For the non-convex 2-dimensional problem $\widetilde{\mathrm{P}}_2$ the minimum value $\widetilde{\mathscr R}(\Omega)$ is found with accuracy $0.61%$. The proof of this estimate is carried out with the use of a result related to the Monge problem of mass transfer, which is also solved in this paper. This problem is as follows: find $\displaystyle\inf_{T\in\mathscr T}\int_\Pi\mathrm{f}(\varphi,\tau;T(\varphi,\tau)) d\mu(\varphi,\tau)$, where $\Pi=[-{\pi}/{2},{\pi}/{2}]\times [0,1]$, $d\mu(\varphi,\tau)=\cos\varphi d\varphi d\tau$, $\mathrm{f}(\varphi,\tau;\varphi',\tau') =1+\cos(\varphi+\varphi')$, and $\mathscr T$ is the set of one-to-one maps of $\Pi$ onto itself preserving the measure $\mu$.
Another problem under study is the minimization of $\overline{\mathscr R}(\Omega) =\displaystyle\int_{S^{d-1}}|\mathscr R_\Omega(v)| dv$. The solution of the convex problem $\overline{\mathrm P} _d^c$ and the estimate for the non-convex 2-dimensional problem $\overline{\mathrm P}_2$ obtained in this paper are the same as for the problems $\widetilde{\mathrm P} _d^c$ and $\widetilde{\mathrm P}_2$.


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English version:
Sbornik: Mathematics, 2004, 195:7, 1017–1037

Bibliographic databases:

UDC: 517.95
MSC: 49J10, 49Q10, 49Q20
Received: 11.11.2003

Citation: A. Yu. Plakhov, “Newton's problem of the body of minimum mean resistance”, Mat. Sb., 195:7 (2004), 105–126; Sb. Math., 195:7 (2004), 1017–1037

Citation in format AMSBIB
\by A.~Yu.~Plakhov
\paper Newton's problem of the body of minimum mean resistance
\jour Mat. Sb.
\yr 2004
\vol 195
\issue 7
\pages 105--126
\jour Sb. Math.
\yr 2004
\vol 195
\issue 7
\pages 1017--1037

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    This publication is cited in the following articles:
    1. V. L. Levin, “Optimality conditions and exact solutions to the two-dimensional Monge–Kantorovich problem”, J. Math. Sci. (N. Y.), 133:4 (2006), 1456–1463  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Plakhov A., Gouveia P.D.F., “Problems of maximal mean resistance on the plane”, Nonlinearity, 20:9 (2007), 2271–2287  crossref  mathscinet  zmath  isi
    3. Plakhov A., “Billiards and Two-Dimensional Problems of Optimal Resistance”, Arch Rational Mech Anal, 194:2 (2009), 349–381  crossref  mathscinet  zmath  adsnasa  isi
    4. Plakhov A., “Billiard scattering on rough sets: two-dimensional case.”, SIAM J. Math. Anal., 40:6 (2009), 2155–2178  crossref  mathscinet  zmath  isi
    5. Gouveia P.D.F., Plakhov A.Yu., Torres D.F.M., “On the two-dimensional rotational body of maximal Newtonian resistance”, J. Math. Sci. (N. Y.), 161:6 (2009), 811–819  crossref  mathscinet  zmath
    6. Gouveia P.D.F., Plakhov A., Torres D.F.. M., “Two-dimensional body of maximum mean resistance”, Appl. Math. Comput., 215:1 (2009), 37–52  crossref  mathscinet  zmath  isi
    7. Buttazzo G., “A Survey on the Newton Problem of Optimal Profiles”, Variational Analysis and Aerospace Engineering, Springer Series in Optimization and Its Applications, 33, 2009, 33–48  crossref  mathscinet  zmath  isi
    8. Plakhov A., “Problems of Minimal and Maximal Aerodynamic Resistance”, Variational Analysis and Aerospace Engineering, Springer Series in Optimization and Its Applications, 33, 2009, 349–365  crossref  mathscinet  zmath  isi
    9. A. Yu. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics”, Russian Math. Surveys, 64:5 (2009), 873–938  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. Plakhov A., Tchemisova T., Gouveia P., “Spinning rough disc moving in a rarefied medium”, Proc. R. Soc. A, 2010  crossref  mathscinet  zmath  isi  elib
    11. A. Plakhov, “Billiards, scattering by rough obstacles, and optimal mass transportation”, J Math Sci, 2012  crossref  mathscinet
    12. McCann R.J., Guillen N., “Five Lectures on Optimal Transportation: Geometry, Regularity and Applications”, Analysis and Geometry of Metric Measure Spaces, CRM Proceedings & Lecture Notes, 56, eds. Dafni G., McCann R., Stancu A., Amer Mathematical Soc, 2013, 145–180  crossref  mathscinet  zmath  isi
    13. Kryzhevich S., “Motion of a Rough Disc in Newtonian Aerodynamics”, Optimization in the Natural Sciences, Communications in Computer and Information Science, 499, eds. Plakhov A., Tchemisova T., Freitas A., Springer-Verlag Berlin, 2015, 3–19  crossref  isi  scopus
    14. Plakhov A. Tchemisova T., “Problems of optimal transportation on the circle and their mechanical applications”, J. Differ. Equ., 262:3, 2 (2017), 2449–2492  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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