Collision Singularity Research Articles

Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space

AbstractIn this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees an enhanced dissipation effect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one sees Taylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay.

Orbit injection of planet-crossing asteroids

ABSTRACT Solar system Centaurs originate in trans-Neptunian space from where planet orbit crossing events inject their orbits inside the giant planets’ domain. Here, we examine this injection process in the three-body problem by studying the orbital evolution of trans-Neptunian asteroids located at Neptune’s collision singularity as a function of the Tisserand invariant, T. Two injection modes are found, one for T > 0.1, or equivalently prograde inclinations far from the planet, where unstable motion dominates injection, and another for T ≤ 0.1, or equivalently polar and retrograde inclinations far from the planet, where stable motion dominates injection. The injection modes are independent of the initial semimajor axis and the dynamical time at the collision singularity. The simulations uncovered a region in the polar corridor where the dynamical time exceeds the Solar system’s age suggesting the possibility of long-lived primordial polar trans-Neptunian reservoirs that supply Centaurs to the giant planets’ domain.

Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition.

A semi-analytical model for coorbital motion

ABSTRACT A globally valid analytically averaged Hamiltonian model for the coorbital motion is hard to construct because the analytical expansions of the disturbing function usually diverge in the quasi-satellite domain that is close to collision singularity. In this paper, an analytically averaged model for the coorbital motion is proposed in case of the circular restricted three-body problem, which can describe properly the transitions that occur at small eccentricities and inclinations, such as the transition between the horseshoe orbit and the quasi-satellite orbit. With the help of the numerical averaging method, numerical experiments are carried out to show the validity and accuracy of the analytically averaged model. The averaged model proposed here can be easily extended to more complicated cases such as the elliptic three-body problem or the planetary three-body problem.

Analytical Study of the Co-orbital Motion in the Circular Restricted Three-body Problem

In the restricted three-body problem (RTBP), if a small body and a planet stably orbit around a central star with almost exactly the same semimajor axis, and thus almost the same mean motion, this phenomenon is called the co-orbital motion, or equivalently, the 1:1 mean motion resonance. The classical expansion of the disturbing function is divergent when the semimajor axis ratio of the small body to the planet is close to unity. Thus, most of the previous studies on the co-orbital dynamics were carried out through numerical integrations or semi-analytical approaches. In this work, we construct an analytical averaged model for the co-orbital motion in the framework of the circular RTBP. This model is valid in the entire coorbital region except in the vicinity of the collision singularity. The results of the analytical averaged model are in good agreement with the numerical averaged model even for moderate eccentricities and inclinations. The analytical model can reproduce the tadpole, horseshoe and quasi-satellite orbits common in the planar problem. Furthermore, the asymmetry of 1:1 resonance and the compound orbits (Icarus 137:293–314) in the general spatial problem can also be obtained from the analytical model.

Near-Collision Dynamics in a Noisy Car-Following Model

We consider a small stochastic perturbation of an optimal velocity car-following model. We give a detailed analysis of behavior near a collision singularity. We show that collision is impossible in a simplified model without noise, and then show that collision is asymptotically unlikely over large time intervals in the presence of small noise, with the large time interval scaling like the square of the reciprocal of the strength of the noise. Our calculations depend on careful boundary-layer analyses.

Self-similar Profiles for Homoenergetic Solutions of the Boltzmann Equation for Non-cutoff Maxwell Molecules

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term which is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Under smallness assumptions on the drift term, we prove that the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p>2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.

Euler integral as a source of chaos in the three–body problem

In this paper we address, from a purely numerical point of view, the question, raised in Pinzari (2019), Pinzari (2020), and partly considered in Pinzari (2020), Di Ruzza et al. (2020), Chen and Pinzari (2021), whether a certain function, referred to as “Euler Integral”, is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the “unperturbed separatrix”, which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in Gierzkiewicz and Zgliczyński (2019), Zgliczynski and Gidea (2004), Wilczak and Zgliczynski (2003).

Total Collisions in the N-Body Shape Space

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

The collision singularity of the Kepler problem with singular perturbations

The collision singularity of the Kepler problem with singular perturbations

Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates

In this article, the existence of mass-conserving solutions is investigated to the continuous coagulation and collisional breakage equation with singular collision kernels. Here, the probability distribution function attains singularity near the origin. The existence result is constructed by using both conservative and non-conservative truncations to the continuous coagulation and collisional breakage equation. The proof of the existence result relies on a classical weak $ L^1 $ compactness method.

A Symmetric Spatial Periodic Orbit in the 2n-Body Problem

For each given $$n \ge 2$$ , we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body problem. Furthermore, with the help of numerical calculation, the orbit is spatial in the equal-mass four-body problem.

Approximation to Singular Quadratic Collision Model in Fokker--Planck--Landau Equation

We propose a Hermite-Galerkin spectral method to numerically solve the spatially homogeneous Fokker-Planck-Landau equation with singular quadratic collision model. To compute the collision model, we adopt a novel approximation formulated by a combination of a simple linear term and a quadratic term very expensive to evaluate. Using the Hermite expansion, the quadratic term is evaluated exactly by calculating the spectral coefficients. To deal with singularities, we make use of Burnett polynomials so that even very singular collision model can be handled smoothly. Numerical examples demonstrate that our method can capture low-order moments with satisfactory accuracy and performance.

Linking low- to high-energy dynamics of invariant manifolds, transit orbits, and singular collision orbits in the planar circular restricted three-body problem

The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.

Application of Morse index in weak force N-body problem

Due to collision singularities, the Lagrange action functional of the N-body problem in general is not differentiable. Because of this, the usual critical point theory cannot be applied to this problem directly. Following ideas from Arioli et al (2006 Commun. Math. Phys. 268 439–63); Bahri and Rabinowitz (1991 Ann. Inst. Henri Poincaré Anal. Non Linéaire 8 561–649); Tanaka (1993 Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 215–38), we introduce a notion called weak critical point for such an action functional, as a generalization of the usual critical point. A corresponding definition of Morse index for such a weak critical point will also be given. Moreover it will be shown that the Morse index gives an upper bound of the number of possible binary collisions in a weak critical point of the N-body problem with weak force potentials including the Newtonian potential.

Integrability of the spatial restricted three-body problem near collisions (an announcement)

We present the integration of the spatial circular restricted three-body problem in a neighbourhood of its collision singularities by extending an idea of Tullio Levi-Civita.

On the Manev spatial isosceles three-body problem

We study the isosceles three-body problem with Manev interaction. Using a McGehee-type technique, we blow up the triple collision singularity into an invariant manifold, called the collision manifold, pasted into the phase space for all energy levels. We find that orbits tending to/ejecting from total collision are present for a large set of angular momenta. We also discover that as the angular momentum is increased, the collision manifold changes its topology.

Embedding the Kepler Problem as a Surface of Revolution

Solutions of the planar Kepler problem with fixed energy h determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on ℝ2 if h ⩾ 0 or on a disk D ⊂ ℝ2 if h < 0. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when h < 0. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in ℝ3 or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with h ⩾ 0 as surfaces of revolution in ℝ3 are constructed explicitly but no such embedding exists for h < 0 due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.

Regularization of the Circular Restricted Three Body Problem on Surfaces of Constant Curvature

We consider a restricted three body problem on surfaces of constant curvature. As in the classical Newtonian case the collision singularities occur when the position particle with infinitesimal mass coincides with the position of one of the primaries. We prove that the singularities due to collision can be locally (each one separately) and globally (both as the same time) regularized through the construction of Levi-Civita and Birkhoff type transformations respectively. As an application we study some general properties of the Hill’s regions and we present some ejection–collision orbits for the symmetrical problem.

Multiple Periodic Orbits Connecting a Collinear Configuration and a Double Isosceles Configuration in the Planar Equal-Mass Four-Body Problem

Abstract By applying our variational method, we show that there exist 24 local action minimizers connecting two prescribed configurations: a collinear configuration and a double isosceles configuration in H 1 ⁢ ( [ 0 , 1 ] , χ ) {H^{1}([0,1],\chi)} in the planar equal-mass four-body problem. Among the 24 local action minimizers, we prove that the one with the smallest action has no collision singularity and it can be extended to a periodic or quasi-periodic orbit. Furthermore, if all the 24 local action minimizers are free of collision, we show that they can generate sixteen different periodic orbits.