The problem of minimal resistance, old and new (2511.01041v1)

Abstract: Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically symmetric bodies, the problem was later extended to general convex shapes, as explored in \cite{BK93}, \cite{BFK95}. Since then, this broader formulation has inspired a number of articles dedicated to the study of the geometric and analytical properties of optimal shapes, with particular attention to their structure, regularity, and behavior under various constraints. In this article, we provide a comprehensive overview of the principal results that have been established, highlighting the main theoretical advancements. Furthermore, we introduce some new directions of research, some of which were described in \cite{P12}, that offer promising perspectives for future investigation.

• The paper presents both classical and extended formulations of the minimal resistance problem, reviewing Newton's original model and its modern generalizations.
• It demonstrates rigorous results on compactness and existence of minimizers within constrained concave function spaces, ensuring well-posedness.
• It reveals symmetry-breaking and regularity limitations of optimal shapes while extending the model to incorporate positive temperature and non-radial designs.

The Problem of Minimal Resistance: Classical Foundations and Modern Developments

Historical Context and Mathematical Formulation

The minimal resistance problem, originating with Newton's 1685 analysis, seeks the optimal shape of a body moving at constant velocity through a fluid, minimizing the resistance encountered. Newton's model, based on an idealized inviscid and incompressible medium composed of non-interacting particles, led to the sine-squared pressure law: the pressure at a surface point is proportional to $\sin^2\vartheta(x)$, where $\vartheta(x)$ is the angle between the surface normal and the flow direction. The resistance functional for a body profile $u$ over a domain $\Omega$ is given by

$$F(u) = \int_\Omega \frac{1}{1 + |\nabla u(x)|^2} \, dx,$$

with the optimization problem posed as minimizing $F(u)$ over admissible $u$.

The classical model assumes only first impacts contribute to resistance, neglecting tangential friction, vorticity, and turbulence. While simplistic, this framework remains relevant for rarefied gases, high-speed flows, and slender bodies.

Well-Posedness and Admissible Classes

The paper rigorously addresses the ill-posedness of the unconstrained problem, demonstrating that without constraints, the infimum of $F(u)$ is zero but unattainable. Even with a height constraint $0 \leq u(x) \leq M$, pathological oscillatory sequences can drive resistance arbitrarily low, violating the single-impact assumption. To ensure existence of minimizers, the admissible class is restricted to nonnegative, concave functions with bounded height:

$$C_M = \{ u : \Omega \to \mathbb{R},\ u \text{ concave},\ 0 \leq u \leq M \}.$$

Compactness in $W^{1,p}_{\text{loc}}(\Omega)$ for $p < \infty$ is established, guaranteeing existence of minimizers for a broad class of functionals, including the Newtonian resistance.

Alternative constraints, such as fixed volume or surface area, are analyzed. For volume,

$$V_V = \{ u \text{ concave},\ u \geq 0,\ \int_\Omega u\,dx \leq V \},$$

and for surface area,

$$H_S = \left\{ u \geq 0,\ u \text{ concave},\ \int_\Omega \sqrt{1 + |\nabla u|^2}\,dx + \int_{\partial\Omega} u\,d\mathcal{H}^{d-1} \leq S \right\}.$$

Geometric estimates relate these classes to $C_M$, ensuring compactness and existence of minimizers.

Regularity and Symmetry Breaking

A key result is the absence of globally regular solutions: at differentiable points, either $\nabla u(x) = 0$ or $|\nabla u(x)| \geq 1$, precluding $C^1$ regularity. The optimal profile for radial domains exhibits a flat region at the top, with explicit parametric solutions derived via the Euler-Lagrange equation. As $M/R \to \infty$, the radius of the flat region and the relative resistance scale as $r_0/R \sim (M/R)^{-3}$ and $C_0 \sim (M/R)^{-2}$, respectively.

Contrary to intuition, the optimal solution for circular domains is not necessarily radial. The paper proves that in regions where $u < M$ and $u$ is $C^2$, the Hessian determinant vanishes, $\det \nabla^2 u \equiv 0$, a flatness condition not satisfied by the radial solution. This symmetry-breaking phenomenon is substantiated by explicit non-radial optimal shapes.

Intrinsic and Generalized Formulations

The intrinsic model reformulates resistance as a boundary integral over the convex body $E$:

$$F(E) = \int_{\partial E} f(x, \nu(x))\,d\mathcal{H}^d(x),$$

where $f$ is a continuous integrand and $\nu(x)$ the unit normal. Existence of minimizers is established for convex bodies between compact sets $K \subset E \subset Q$, with volume or surface constraints as needed.

Positive Temperature and Extended Models

The classical model's neglect of sub-maximal regions and thermal motion is addressed by considering fluids at positive temperature, where particles have a velocity distribution $\rho(v)$. The resistance force becomes

$$R(E) = \int_{\partial E} f(\nu(x))\,d\mathcal{H}^d,$$

with

$$f(\nu) = -2\nu \int_{\mathbb{R}^d} (v \cdot \nu)_-^2 \rho(v)\,dv.$$

This generalization incorporates the full geometry of $E$ and the statistical mechanics of the medium.

Related Problems and Research Directions

The paper surveys several extensions:

• Relaxing convexity to allow bodies with at most single impacts per particle, broadening admissible shapes.
• Accounting for multiple impacts, requiring billiard-theoretic analysis.
• Incorporating tangential friction, relevant for real-world applications.
• Analyzing rotating bodies and the Magnus effect, with implications for aerodynamics.
• Studying low visibility and retroreflector problems, connecting resistance minimization to optical properties.

Conclusion

This work provides a comprehensive synthesis of the minimal resistance problem, from Newton's classical formulation to modern generalizations. The rigorous analysis of well-posedness, regularity, and symmetry breaking advances the theoretical understanding of shape optimization under physical constraints. The intrinsic and positive temperature models extend applicability to more realistic scenarios. The surveyed related problems highlight the rich interplay between variational analysis, geometry, and physical modeling, suggesting fruitful directions for future research in mathematical physics, optimization, and applied mathematics.