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

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 $\widetilde{\mathrm P}_2$.

DOI: https://doi.org/10.4213/sm836

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

Bibliographic databases:

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

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
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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
2. Plakhov A., Gouveia P.D.F., “Problems of maximal mean resistance on the plane”, Nonlinearity, 20:9 (2007), 2271–2287
3. Plakhov A., “Billiards and Two-Dimensional Problems of Optimal Resistance”, Arch Rational Mech Anal, 194:2 (2009), 349–381
4. Plakhov A., “Billiard scattering on rough sets: two-dimensional case.”, SIAM J. Math. Anal., 40:6 (2009), 2155–2178
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
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
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
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
9. A. Yu. Plakhov, “Scattering in billiards and problems of Newtonian aerodynamics”, Russian Math. Surveys, 64:5 (2009), 873–938
10. Plakhov A., Tchemisova T., Gouveia P., “Spinning rough disc moving in a rarefied medium”, Proc. R. Soc. A, 2010
11. A. Plakhov, “Billiards, scattering by rough obstacles, and optimal mass transportation”, J Math Sci, 2012
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
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
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
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