AI 中文总结
本文证明Vlasov方程在d≥4维、0<α<3范围内对吸引和排斥幂律势的全局适定性，无需初值小性条件，并利用维里恒等式和补偿可积性不等式处理吸引情形下的有限时间破裂。
AI 中文摘要
我们研究具有一般幂律（Riesz型）势（指数为\\(\alpha\\)）的Vlasov方程。我们证明在任意维度\\(d\ge4\\)中，对于吸引和排斥相互作用，在整个范围\\(0<\alpha<3\\)内，对任意非负紧支撑有界初值，全局适定性成立。对初值不施加小性条件。对于吸引问题，当\\(\alpha\ge3\\)时，维里恒等式给出负能量解的有限时间破裂。全局适定性的证明结合了特征线分析、拉格朗日坐标下的估计，以及Denis Serre提出的\\(补偿可积性\\)不等式的一个关键应用。
英文摘要
We study the Vlasov equation with general power-law (Riesz-type) potentials with exponent \(α\). We prove global well-posedness in every dimension \(d\ge4\), for both attractive and repulsive interactions, throughout the range \(0<α<3\), for arbitrary nonnegative compactly supported bounded initial data. No smallness condition is imposed on the initial data. For the attractive problem, the virial identity gives finite-time breakdown for negative-energy solutions when \(α\ge3\). The proof of global well-posedness combines the analysis of characteristics, estimates in Lagrangian coordinates, and a crucial application of a \emph{compensated integrability} inequality due to Denis Serre.
