Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem

Pages 499-550 from Volume 192 (2020), Issue 2 by Ezequiel Maderna, Andrea Venturelli

Abstract

We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal’s Theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

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Keywords
, ,
Mathematical Subject Classification
Primary: , Secondary: ,
DOI
https://doi.org/10.4007/annals.2020.192.2.5
MR
4151083
Milestones
Received: 8 October 2019
Revised: 27 May 2020
Accepted: 12 June 2020
Published online: 9 September 2020

Authors

Ezequiel Maderna

IMERL & CMAT, Universidad de la República, Montevideo, Uruguay

Andrea Venturelli

Laboratoire de Mathématiques d'Avignon, Avignon, France