Resumen de Global Existence and Singularity of the N-Body Problem with Strong Force

Yanxia Deng, Slim Ibrahim

• We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrödinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem 4, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function Kω. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For N≥3, we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of Kω.