A common structure for modified scattering: Vlasov--Riesz, Hartree, and their coupling
Abstract
We identify a common action--density structure for the Hartree and Vlasov--Riesz equations and their coupling. Along free rays, the feedback between nonlinear actions and rescaled densities governs wave phase modulations and kinetic momentum translations. This structure persists under the exchange of sources, providing convergent modified profiles and an explicit recursive construction of the asymptotic corrections. We thereby establish small-data global existence and modified scattering in three spatial dimensions for the Hartree equation, the Vlasov--Riesz equation, and the coupled Vlasov--Hartree system, with interaction potential λ|x|−α, λ∈ℝ and 0<α≤ 1/2. In this strongly long-range regime, the asymptotic action contains finitely many successive corrections beyond its leading term. To our knowledge, these are the first forward modified-scattering results for all three models throughout this range.
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BibTeX
@misc{huang2026common,
title = {{A common structure for modified scattering: Vlasov--Riesz, Hartree, and their coupling}},
author = {Wenrui Huang and Mengyi Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02643},
eprint = {2610.02643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02643},
}
RIS
TY - UNPB
TI - A common structure for modified scattering: Vlasov--Riesz, Hartree, and their coupling
AU - Wenrui Huang
AU - Mengyi Xie
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02643
UR - https://arxiv.org/abs/2610.02643
AB - We identify a common action--density structure for the Hartree and Vlasov--Riesz equations and their coupling. Along free rays, the feedback between nonlinear actions and rescaled densities governs wave phase modulations and kinetic momentum translations. This structure persists under the exchange of sources, providing convergent modified profiles and an explicit recursive construction of the asymptotic corrections. We thereby establish small-data global existence and modified scattering in three spatial dimensions for the Hartree equation, the Vlasov--Riesz equation, and the coupled Vlasov--Hartree system, with interaction potential λ|x|^−α, λ∈ℝ and 0<α≤ 1/2. In this strongly long-range regime, the asymptotic action contains finitely many successive corrections beyond its leading term. To our knowledge, these are the first forward modified-scattering results for all three models throughout this range.
ER -