Newton's problem of minimal resistance asks for the shape of a body moving through a rarefied medium of non-interacting particles that minimizes the drag produced by elastic reflections. For two-dimensional rough bodies, Plakhov's 2009 theory assigns to each body a billiard-generated scattering measure and reduces minimal and maximal mean resistance, for bodies translating and slowly rotating, to Monge-Kantorovich optimal transport problems. The multidimensional characterization and resistance optimization promised in that paper were later resolved by the author's monograph, and the slow-rotation and slow-oscillation programs were carried through to explicit solutions. What remains open is the regime the original paper explicitly postponed outside its hypotheses: bodies whose rotation rate is comparable to the translational speed (the non-quasi-static, or fast-rotation, regime), where no characterization of admissible scattering laws and no min/max resistance theory exists; thermal (positive-temperature) media beyond convex axisymmetric bodies; and the further reduction program for resistance functionals whose cost functions are determined by particular physical problems.

Background

In Newton's classical problem of the body of minimal resistance, a body translates at constant velocity through a rarefied medium, meaning a gas of particles so dilute that particle-particle interactions are neglected, and each particle meeting the body reflects elastically (angle of incidence equals angle of reflection). The resistance is the mean drag exerted by the reflected particles, and one seeks the body shape minimizing it. Newton (1687) solved the axisymmetric convex case; the modern theory allows nonconvex and 'rough' bodies, meaning bodies whose boundaries are endowed with fine-scale billiard microstructure that redirects incident particles according to an effective scattering law. The key device, introduced for planar bodies in Plakhov's 2009 paper 'Billiards and Two-Dimensional Problems of Optimal Resistance', is the scattering measure: to each compact connected set B⊂R2B\subset\mathbb{R}^2 one associates a measure νB\nu_B on ∂(conv B)×[−π/2,π/2]\partial(\mathrm{conv}\,B)\times[-\pi/2,\pi/2] recording how particles entering at a given point and angle are redistributed in outgoing angle. The paper characterized exactly which measures arise from billiards (via a measure-preservation/adjointness condition and a density theorem), and showed that minimizing or maximizing the mean resistance of a translating or slowly rotating body reduces to a Monge-Kantorovich optimal transport problem, that is, the classical problem of transporting one measure to another at least cost, with a cost function determined by the physics: for the resistance cost one obtains the cost c(φ,φ+)=1+cos⁡(φ−φ+)c(\varphi,\varphi^+)=1+\cos(\varphi-\varphi^+) between incoming and outgoing angles. In the slow-rotation (quasi-static) regime the body rotates so slowly that each particle-body collision occurs with the body effectively frozen, so the time-averaged resistance is an integral of static resistances weighted by the rotation law; uniform slow rotation gives the factor 0.9878…0.9878\ldots relative to the unroughened value, from a one-dimensional transport problem on the circle, and the supremum over roughenings is 1.5 times the unroughened value. The paper's program discussion of open directions (accessible author preprint arXiv:math/0607129, end of Section 2.2, together with the Example 2 remark on C0C^0-but-not-C1C^1 convergence of approximating boundaries) comprised: (a) extension of the characterization and density theorems to R3\mathbb{R}^3 and higher dimensions; (b) motion and medium states beyond the hypotheses, namely non-uniform rotations, rotation rates comparable to particle velocities, and thermally (chaotically) moving media; (c) regularity and constructive description of extremals, the optimal values in the solved examples being attained only as limits of bodies whose boundaries converge in C0C^0 but not in C1C^1; and (d) the program of reducing other resistance functionals — each with the cost function determined by the particular physical problem under consideration — to solvable Monge-Kantorovich problems.

Problem Statement

Develop a theory of minimal and maximal mean resistance for two-dimensional rough bodies in the non-quasi-static (fast) rotation regime, where the tangential velocity of the body's surface due to rotation is comparable to the relative velocity of the medium's particles, so that successive particle-body interactions occur at appreciably different body orientations and the 'frozen body' reduction to a static Monge-Kantorovich problem on the circle no longer applies. Concretely, for a body in a rarefied medium of non-interacting particles with elastic reflections and a prescribed rotation law of finite rate: (1) characterize the admissible time-dependent scattering laws realizable by billiard microstructure on a rotating body, in the spirit of the 2009 characterization of the measures νB\nu_B on ∂(conv B)×[−π/2,π/2]\partial(\mathrm{conv}\,B)\times[-\pi/2,\pi/2] for the static case; (2) reduce, or show it is impossible to reduce, the min/max of the time-averaged mean resistance over roughenings to a solvable transport-type problem; and (3) determine whether optimal roughenings are attained by genuine bodies or only by measure-valued limits — in the solved quasi-static cases the optimal values are attained only by sequences of bodies whose boundaries converge in C0C^0 but not in C1C^1 — and (4) continue the source program of reducing further resistance functionals, each with the cost function determined by the particular physical problem, to solvable transport-type problems beyond the exactly solvable cases of the slow-rotation and slow-oscillation programs. Included in the same postponed program: extend the thermal-medium (positive-temperature) theory, concerning media whose particles carry a chaotic thermal velocity distribution rather than a common drift velocity, beyond convex axisymmetric translating bodies. This is the part of the source paper's own open-directions program that survived the 2012 monograph and the 2017-2026 slow-oscillation developments; no answer or partial theory beyond special geometries analyzed case by case (spinning disc, rotating rod and ball, and similar special systems) is known. Sub-questions (1)-(3) constitute the fast-rotation part of the requested theory, while (4) and the thermal-medium extension belong to the same postponed program and stand on their own. Establishing or refuting any single sub-question, including (4) or the thermal-medium extension, is itself a self-contained result.

References

  1. Alexander Plakhov, "Newton's problem of the body of minimum mean resistance", Sbornik: Mathematics 195(7):1017-1037, 2004. DOI 10.1070/sm2004v195n07abeh000836, https://doi.org/10.1070/sm2004v195n07abeh000836
  2. Alexander Plakhov and D. F. M. Torres, "Newton's aerodynamic problem in media of chaotically moving particles", Sbornik: Mathematics 196(6):885-933, 2005. DOI 10.1070/sm2005v196n06abeh000904, https://doi.org/10.1070/sm2005v196n06abeh000904
  3. Alexander Plakhov, "Billiards and Two-Dimensional Problems of Optimal Resistance", Archive for Rational Mechanics and Analysis 194(2):349-381, 2009. DOI 10.1007/s00205-008-0137-1, https://doi.org/10.1007/s00205-008-0137-1 (source paper)
  4. Alexander Plakhov, "Billiard scattering on rough sets: two-dimensional case", arXiv:0711.0616, 2007. https://arxiv.org/abs/0711.0616
  5. Alexander Plakhov, "Scattering in billiards and problems of Newtonian aerodynamics", Russian Mathematical Surveys 64(5):873-938, 2009. DOI 10.1070/RM2009V064N05ABEH004642, https://doi.org/10.1070/RM2009V064N05ABEH004642
  6. Alena Aleksenko and Alexander Plakhov, "Bodies of zero resistance and bodies invisible in one direction", Nonlinearity 22(6):1247-1258, 2009. DOI 10.1088/0951-7715/22/6/001, https://doi.org/10.1088/0951-7715/22/6/001; also arXiv:0809.0108, https://arxiv.org/abs/0809.0108
  7. Alexander Plakhov, Tatiana Tchemisova, and Paulo Gouveia, "Spinning rough disc moving in a rarefied medium", Proceedings of the Royal Society A 466(2119):2033-2055, 2010. DOI 10.1098/rspa.2009.0518, https://doi.org/10.1098/rspa.2009.0518
  8. Alexander Plakhov, "Exterior Billiards", Springer, 2012. ISBN 978-1-4614-4480-0, DOI 10.1007/978-1-4614-4481-7, https://doi.org/10.1007/978-1-4614-4481-7 (monograph)
  9. Alexander Plakhov, "Billiards, scattering by rough obstacles, and optimal mass transportation", Journal of Mathematical Sciences 182(2):233-245, 2012. DOI 10.1007/s10958-012-0744-0, https://doi.org/10.1007/s10958-012-0744-0
  10. Alexander Plakhov and Tatiana Tchemisova, "Problems of optimal transportation on the circle and their mechanical applications", Journal of Differential Equations 262(3):2449-2492, 2017. DOI 10.1016/j.jde.2016.10.049, https://doi.org/10.1016/j.jde.2016.10.049
  11. Sergey Kryzhevich and Alexander Plakhov, "Rotating rod and ball", Journal of Mathematical Analysis and Applications 533:128018, 2024. DOI 10.1016/j.jmaa.2023.128018, https://doi.org/10.1016/j.jmaa.2023.128018
  12. Peter Rudzis, "Rough Collisions", Memoirs of the American Mathematical Society 304(1533), 2024. DOI 10.1090/memo/1533, https://doi.org/10.1090/memo/1533; also arXiv:2203.09102, https://arxiv.org/abs/2203.09102
  13. Giuseppe Buttazzo, "The problem of minimal resistance, old and new", arXiv:2511.01041, 2025. https://arxiv.org/abs/2511.01041
  14. Roman Karasev, Anton Petrunin, and Alexander Plakhov, "An isoperimetric inequality for mean shadow", arXiv:2606.10608, 2026. https://arxiv.org/abs/2606.10608
  15. Lev Lokutsievskiy and Alexander Plakhov, "Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance", arXiv:2608.17783, 2026. https://arxiv.org/abs/2608.17783
  16. Alexander Plakhov, "Notes on Invisibility and Zero Resistance in Billiards", Proceedings of the Steklov Institute of Mathematics 332(1):152-160, 2026. DOI 10.1134/s0081543826600481, https://doi.org/10.1134/s0081543826600481