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

摘要

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrodinger equations) to characterize solutions in theN-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 belowthe excited energy in Theorem 4, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function K.. 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 K-omega.