论文标题
无需压力的欧拉对准系统的质量浓度集的几何结构
Geometric Structure of Mass Concentration Sets for Pressureless Euler Alignment Systems
论文作者
论文摘要
我们研究了Euler对齐系统的限制动力学，具有光滑，重尾相互作用的内核$ ϕ $和单向速度$ \ mathbf {u} =（u，0，\ ldots，0）$。我们演示了熵函数$ e_0 = \ partial_1 u_0 + ϕ*ρ_0$与限制的“浓度集”，即，限制密度度量的单数部分的支持。在典型的情况下，羊群会汇总$ c^1 $ hypersurfaces的联合：在限制流量图下$ e_0 $的零集的图像。这种对应关系还使我们能够对与限制动力学相关的良好属性发表陈述，包括浓度集的尺寸上的锋利上限，仅取决于$ e_0 $的平滑度。为了促进和背景化我们对限制密度度量的分析，我们还包括对欧拉对准系统的良好性，羊群和稳定性的说明性讨论，其中大多数是新的。
We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel $ϕ$ and unidirectional velocity $\mathbf{u} = (u, 0, \ldots, 0)$. We demonstrate a striking correspondence between the entropy function $e_0 = \partial_1 u_0 + ϕ*ρ_0$ and the limiting 'concentration set', i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of $C^1$ hypersurfaces: the image of the zero set of $e_0$ under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of $e_0$. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.
