Geometric structure of mass concentration sets for pressureless Euler alignment systems
Geometric structure of mass concentration sets for pressureless Euler alignment systems
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DOI：
10.1016/j.aim.2022.108290
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发表时间：
2020-08
影响因子：
1.7
通讯作者：
Daniel Lear;T. Leslie;R. Shvydkoy;E. Tadmor
中科院分区：
文献类型：
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作者：
Daniel Lear;T. Leslie;R. Shvydkoy;E. Tadmor
研究了具有光滑重尾相互作用核ϕ和单向速度u=(u，0，…)的欧拉对准系统的极限动力学，0)。我们证明了熵函数e0=∂1 u0+ϕ⁎ρ0与极限‘浓度集’之间的显著对应，即极限密度测度的奇异部分的支撑性。在一个典型的场景中，羊群朝着C1超曲面的并集经历聚集：极限流图下e0的零集的图像。这种对应关系也使我们能够陈述与极限动力学有关的精细性质，包括仅取决于e0的光滑度的浓度集的维度的尖锐上界。为了便于我们对极限密度度量的分析，我们还包括了对欧拉排列系统的适定性、聚集性和稳定性的说明性讨论，其中大部分是新的。
We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel ϕ and unidirectional velocity u=(u, 0,…, 0). We demonstrate a striking correspondence between the entropy function e 0=∂ 1 u 0+ ϕ⁎ ρ 0 and the limiting ‘concentration set’, ie, 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.