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Multisoliton solutions and blow up for the $L^2$-critical Hartree equation

Jaime Gómez, Tobias Schmid, Yutong Wu

TL;DR

The paper addresses the long-time and blow-up dynamics of the $L^2$-critical Hartree equation in four dimensions by constructing multisoliton solutions whose cores follow an $m$-body Newton-type law with inverse-square interactions. The authors extend the Krieger–Martel–Raphaël and Wu framework to a nonlocal Hartree setting, using a refined approximation and modulation strategy to align the multi-soliton entities with hyperbolic or parabolic trajectories and to control degeneracies in generalized root spaces. Through a pseudo-conformal transformation, they derive finite-time blow-up at multiple points (or a single point) with the sharp rate $\|\nabla u(t)\|_{L^2} \sim |t|^{-1}$, and provide precise modulation/evolution estimates that close a bootstrap argument. This work connects $m$-body dynamics with nonlocal dispersive PDE behavior, offering new insights into multisoliton interactions and blow-up mechanisms in critical Hartree-type equations.

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. 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, 2009] and the third author's recent extension. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.

Multisoliton solutions and blow up for the $L^2$-critical Hartree equation

TL;DR

The paper addresses the long-time and blow-up dynamics of the -critical Hartree equation in four dimensions by constructing multisoliton solutions whose cores follow an -body Newton-type law with inverse-square interactions. The authors extend the Krieger–Martel–Raphaël and Wu framework to a nonlocal Hartree setting, using a refined approximation and modulation strategy to align the multi-soliton entities with hyperbolic or parabolic trajectories and to control degeneracies in generalized root spaces. Through a pseudo-conformal transformation, they derive finite-time blow-up at multiple points (or a single point) with the sharp rate , and provide precise modulation/evolution estimates that close a bootstrap argument. This work connects -body dynamics with nonlocal dispersive PDE behavior, offering new insights into multisoliton interactions and blow-up mechanisms in critical Hartree-type equations.

Abstract

We construct multisoliton solutions for the -critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. Precisely, we consider the -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 distinct points in the former case. The approach we take is based on the ideas of [Krieger-Martel-Raphaël, 2009] and the third author's recent extension. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.

Paper Structure

This paper contains 12 sections, 29 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. $m$-body dynamics
  4. Statement of the results
  5. Construction of approximate solutions
  6. The trajectories
  7. Bootstrap reduction
  8. Modulation and error control
  9. Modulation
  10. Error estimate
  11. Auxiliary details for the non-degeneracy
  12. Proof of Propositions \ref{['hyp']} and \ref{['parab']}

Key Result

Lemma 1.2

Let $\alpha(t)$ be a global solution of m-body. Then $\max_{j < k}|\alpha_{j k}(t)| = O(t)$ as $t \to \infty$ and $\liminf_{t \to \infty} a(t) > 0$. Further for some $v \in \mathcal{X}$ we have $\alpha(t) = v t + O(t^{\frac{1}{2}})$ as $t \to \infty$. The vector $v = \lim_{t \to \infty} \frac{\alpha

Theorems & Definitions (60)

  • Remark 1.1
  • Lemma 1.2
  • Definition 1.3
  • Proposition 1.4: hyperbolic solutions
  • Proposition 1.5: parabolic solutions
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8: Finite-time blow up at distinct points
  • Corollary 1.9: Finite-time collision blow up
  • Remark 1.10
  • ...and 50 more