Recent zbMATH articles in MSC 70H20https://www.zbmath.org/atom/cc/70H202022-01-14T13:23:02.489162ZWerkzeugViscosity solutions and hyperbolic motions: a new PDE method for the \(N\)-body problemhttps://www.zbmath.org/1475.700182022-01-14T13:23:02.489162Z"Maderna, Ezequiel"https://www.zbmath.org/authors/?q=ai:maderna.ezequiel"Venturelli, Andrea"https://www.zbmath.org/authors/?q=ai:venturelli.andreaThis long article is about the asymptotic behaviour of the Newtonian \(N\)-body problem in dimension at least two. Hyperbolic motions, following Chazy, are those such that each body has a different limit velocity vector \(a\), as time is going to \(+\infty\). It follows that there are no singularities in the future. The only known motions of this kind are of homographic type and the configuration is always central. In this article it is proved that any configuration without collisions is the limit shape of some hyperbolic motion, for any choice of the initial positions and positive energy (Theorem 1.1), including collisions in the past. In geometric terms: for any positive energy, initial condition \(p(0)\) and any configuration \(a\) without collision, there is a geodesic ray of the Jacobi-Maupertuis metric of the \(N\)-body system with asymptotic direction \(a\) and starting at \(p(0)\). The proof makes use of global viscosity solutions of the Hamilton-Jacobi equation associated with the \(N\)-body system. For a given Hamilton-Jacobi equation \(H=h\), a real-valued function \(u\in \mathcal{C}^1(E^N)\), where \(E^N\) is the configuration space, is a viscosity subsolution or supersolution of \(H=h\) if, for any \(\psi\in \mathcal{C}^1(E^N)\), whenever \(u-\psi\) has a local maximum or minimum, then \(H\leq h\) or \(H\geq h\) respectively. When \(u\) is both sub and super solution, then it is a viscosity solution. It is shown, by application of Marchal's theorem [\textit{C. Marchal}, Celest. Mech. Dyn. Astron. 83, No. 1--4, 325--353 (2002; Zbl 1073.70011)], that viscosity subsolutions of the H-J equation coincide with a class of functions (dominated functions), controlled by the Lagrangian action of the \(N\)-body system, and that, under some further condition, viscosity subsolutions are actually viscosity solutions. In addition, it is shown that they are fixed points of the quotient Lax-Oleinik semigroup. The existence of global viscosity solutions is proved by showing that these are actually the set of continuous functions \(u\) on the configuration space, called horofunctions, defined for any configuration \(x\) by
\[
u(x)=\lim _{n\rightarrow \infty} \phi_h(x.p_n)-\phi_h(0,p_n),
\]
for a sequence of configurations \(p_n\) with \(||p_n||\rightarrow +\infty\), where \(\phi_h(a,b)\) is the supercritical action potential for the value \(h>0\) of the H-J equation on curves joining configurations \(a\) and \(b\). A procedure inspired by Gromov's construction of the ideal boundary of a complete locally compact metric space. Thanks to Marchal's theorem, it is shown that these curves are motions of the \(N\)-body system. The article is largely self-contained and the matter is clearly exposed and discussed into details. Several open problems are considered at the conclusion.
Reviewer: Giovanni Rastelli (Vercelli)Wormhole cosmic censorship: an analytical proofhttps://www.zbmath.org/1475.831052022-01-14T13:23:02.489162Z"Del Águila, Juan Carlos"https://www.zbmath.org/authors/?q=ai:del-aguila.juan-carlos"Matos, Tonatiuh"https://www.zbmath.org/authors/?q=ai:matos.tonatiuh