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September 2020 Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem
Ezequiel Maderna, Andrea Venturelli
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Ann. of Math. (2) 192(2): 499-550 (September 2020). DOI: 10.4007/annals.2020.192.2.5

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 $H(x,d_x u)=h$. 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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Ezequiel Maderna. Andrea Venturelli. "Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem." Ann. of Math. (2) 192 (2) 499 - 550, September 2020. https://doi.org/10.4007/annals.2020.192.2.5

Information

Published: September 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.2.5

Subjects:
Primary: 70F10 , 70H20
Secondary: 37J50 , 49L25

Keywords: $n$-body problem , Hamilton-Jacobi equation , viscosity solutions

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 2 • September 2020
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