Multisoliton Solutions and Blow Up for the \(L^2\)-Critical Hartree Equation

Abstract

We construct multisoliton solutions for the \(L^2\)-critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. More precisely, we consider the m-body hyperbolic and parabolic non-trapped dynamics. The pseudo-conformal symmetry then implies finite-time collision blow-up in the latter case and a solution blowing up at m distinct points in the former case. The approach we take is based on the ideas of Krieger–Martel–Raphaël [10] and the third author’s recent extension [36]. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.