发表机构
Brown University; Yale University（布朗大学; 耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发现Hartree、Vlasov--Riesz及其耦合方程共有的作用-密度结构，利用该结构在三维空间中建立了小初值全局存在性和修正散射，覆盖强长程势$\lambda|x|^{-\alpha}$（$0<\alpha\leq1/2$），并给出首个前向修正散射结果。
AI 中文摘要
我们识别出Hartree方程、Vlasov--Riesz方程及其耦合方程所共有的作用-密度结构。沿自由射线，非线性作用与重标度密度之间的反馈控制着波相位调制和动力学动量平移。该结构在源交换下保持不变，从而提供收敛的修正轮廓以及渐近修正的显式递归构造。由此，我们建立了在三维空间中，对于Hartree方程、Vlasov--Riesz方程以及耦合的Vlasov--Hartree系统，在相互作用势为$\lambda|x|^{-\alpha}$（其中$\lambda\in\mathbb R$，$0<\alpha\leq 1/2$）时的小初值全局存在性和修正散射。在这个强长程区域中，渐近作用包含其首项之外的有限多个连续修正。据我们所知，这是针对所有这三个模型在该范围内首次得到的前向修正散射结果。
英文摘要
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 $λ|x|^{-α}$, $λ\in\mathbb R$ and $0<α\leq 1/2$. 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.
