The Newton's problem assuming non-constant density of the fluid

Technical Summary: The Newton's Problem of Minimal Resistance with Non-Constant Density

Problem Statement
This paper investigates a variant of Newton's classical problem of minimal resistance, which seeks the shape of a solid body moving at constant velocity through a fluid that offers the least resistance. While the classical model assumes a homogeneous fluid, this study addresses the more realistic scenario where the fluid density decreases exponentially with altitude, a condition characteristic of the Earth's atmosphere. The density is modeled as $\rho(z) = \rho_0 e^{-cz}$, where $z$ represents altitude. The body moves along the $z$-coordinate, and the resistance functional is derived based on the sine-squared pressure law, assuming no friction and that particles do not re-collide with the surface after impact.

The resulting resistance functional for a surface defined by a graph $u: \Omega \to \mathbb{R}$ is given by:
$$E[u] = \int_{\Omega} \frac{e^{-u}}{1 + |Du|^2} \, dxdy$$
Unlike the classical Newton functional, the presence of the exponential weight $e^{-u}$ introduces a significant structural change in the associated Euler-Lagrange equation. The paper explicitly states that its focus is on the existence and regularity of classical $C^2$ extremals (radial solutions) rather than the determination of a global minimizer, noting that the unconstrained minimum resistance problem in the classical setting often lacks $C^2$ regularity.

Methodology
The analysis focuses on radial solutions $u(r)$, where the surface is a revolution about the $z$-axis. The Euler-Lagrange equation for the functional $E[u]$ reduces to a second-order nonlinear ordinary differential equation (ODE):
$$r(1 + 4u'^2 + 3u'^4) - 2u'(1 + u'^2) + 2ru''(3u'^2 - 1) = 0$$
The study addresses the initial value problem with conditions $u(0) = 0$ and $u'(0) = 0$. A primary challenge is the singularity of the equation at $r=0$, which prevents the direct application of classical ODE existence theorems.

To overcome this, the author employs a fixed-point argument based on the Banach Fixed Point Theorem. The ODE is reformulated into an integral equation involving the inverse of a specific monotone function $f(x) = x/(1+x^2)^2$. By defining an operator $T$ on a closed ball in the Banach space $C^1([0, R])$, the author proves that $T$ is a contraction for a sufficiently small $R$, thereby establishing the local existence of a $C^2$ solution.

To analyze the global behavior and the maximal domain of the solution, the problem is transformed into a phase plane analysis. The curve is reparametrized by arc length $s$, introducing an angle $\theta(s)$ such that $u'(r) = \tan \theta$. This converts the second-order ODE into an autonomous system of first-order equations for the variables $x(s) = r$ and $\theta(s)$. The behavior of the solutions is studied by examining the trajectories in the phase plane $\Theta$, specifically looking at equilibrium points and the flow near critical lines.

Key Contributions and Results

1. Local Existence and Regularity: The paper proves (Theorem 2.1) that the initial value problem with $u(0)=u'(0)=0$ admits a solution $u \in C^2([0, R])$ for some $R > 0$. A crucial finding is that the solution intersects the rotation axis orthogonally ($u'(0)=0$). This contrasts sharply with the classical constant-density model, where radial solutions cannot reach the axis orthogonally unless the solution is a trivial constant function. The proof further establishes that $u''(0) = 1/4$.

2. Monotonicity and Convexity: The analysis demonstrates that the slope $u'(r)$ is strictly positive and increasing in the maximal domain of the solution. Consequently, the radial solution $u(r)$ is convex.

3. Finite Maximal Domain and Critical Slope: Theorem 2.2 establishes that the maximal domain of the solution is a finite interval $[0, r_M)$. The solution terminates at a point where the coefficient of $u''$ in the ODE vanishes, corresponding to a critical slope:
   $$\lim_{r \to r_M} u'(r) = \frac{1}{\sqrt{3}}$$
   Numerical calculations indicate $r_M \approx 1.230$ and $u(r_M) \approx 0.228$. The phase plane analysis reveals that the trajectories approach the line $\theta = \pi/6$ (where $\tan \theta = 1/\sqrt{3}$) asymptotically, and the domain ends at a finite radius.

4. Phase Plane Structure: The study characterizes the phase plane of the associated autonomous system, identifying equilibrium points and showing that the flow around specific points consists of topological circles. It is shown that trajectories starting from the origin terminate at the horizontal line $\theta = \pi/6$ with a finite $x$-coordinate.

Significance and Claims
The paper claims that the exponential decay of fluid density fundamentally alters the geometry of minimal resistance bodies compared to the classical homogeneous model. Specifically, the existence of smooth ($C^2$) radial solutions that intersect the symmetry axis orthogonally is a novel result. In the classical model, such smoothness at the axis is impossible for non-trivial solutions, which typically require a "flat-front" disk (a discontinuity in the derivative) to minimize resistance.

The author posits that these smooth radial solutions provide a theoretical basis for optimal shapes in stratified media. The geometric constraint that the slope is bounded by $1/\sqrt{3}$ at the edge of the domain is highlighted as a significant feature for the design of bodies traversing the atmosphere. The paper concludes that while these $C^2$ extremals exist and are regular, the question of whether they represent global minimizers versus the classical non-smooth minimizers remains an open problem for future research. The work is presented as a rigorous mathematical investigation into the existence and properties of these specific extremals, without proposing new experimental applications or claiming immediate engineering implementation.