论文标题
不可避免的奇异对准的单在
Inevitable monokineticity of strongly singular alignment
论文作者
论文摘要
我们证明，某些类型的度量值映射是单基因的，即速度的分布集中在狄拉克质量中。其中包括针对强烈奇异的cucker-smale模型的弱度量值解决方案，其奇异性$α$或等于环境空间的尺寸。因此，我们能够回答与奇异的cucker-smale模型有关的几个开放问题。首先，我们证明，在非常温和的假设下，它们均匀地紧凑且及时弱连续，对强奇异的cucker-smale动力学方程的弱度量值解决方案是单基因的。这可以解释为从动力学cucker-smale方程中宏观分数Euler对准系统的严格推导，而无需执行任何流体动力学极限。 这表明宏观框架描述了强烈奇异的cucker-smale动力学的巨大界限。 其次，我们执行从Cucker-Sale-Sale粒子系统到分数Euler对齐模型的直接微观平均场极限。这将导致最终结果 - 存在薄弱的Euler对齐系统的弱解决方案，其中几乎是任意的初始数据，其中包括$ \ Mathbb {r}^1 $，包括真空的可能性。在先验假设下，存在均值限制的密度没有原子，就可以将存在扩展到$ \ mathbb {r}^2 $。
We prove that certain types of measure-valued mappings are monokinetic i.e. the distribution of velocity is concentrated in a Dirac mass. These include weak measure-valued solutions to the strongly singular Cucker-Smale model with singularity of order $α$ greater or equal to the dimension of the ambient space. Consequently, we are able to answer a couple of open questions related to the singular Cucker-Smale model. First, we prove that weak measure-valued solutions to the strongly singular Cucker-Smale kinetic equation are monokinetic, under very mild assumptions that they are uniformly compactly supported and weakly continuous in time. This can be interpreted as a rigorous derivation of the macroscopic fractional Euler-alignment system from kinetic Cucker-Smale equation without the need to perform any hydrodynamical limit. This suggests superior suitability of the macroscopic framework to describe large-crowd limits of strongly singular Cucker-Smale dynamics. Second, we perform a direct micro- to macroscopic mean-field limit from the Cucker-Smale particle system to the fractional Euler-alignment model. This leads to the final result -- existence of weak solutions to the fractional Euler-alignment system with almost arbitrary initial data in $\mathbb{R}^1$, including the possibility of vacuum. Existence can be extended to $\mathbb{R}^2$ under the a priori assumption that the density of the mean-field limit has no atoms.
