Critical thresholds in flocking hydrodynamics with non-local alignment
Critical thresholds in flocking hydrodynamics with non-local alignment
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DOI：
10.1098/rsta.2013.0401
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发表时间：
2014-11-13
期刊：
影响因子：
5
通讯作者：
Tan, Changhui
中科院分区：
文献类型：
--
作者：
Tadmor, Eitan;Tan, Changhui
我们研究了欧拉系统的大时间行为与非局部对齐增强。这样的系统出现的流体动力学描述的代理为基础的模型自组织动力学，如。G. Cucker & Smale（2007 IEEE Trans. Autom.控制52，852-862。(doi：10.1109/TAC.2007.895842））和Motsch & Tadmor（2011 J. Stat. Phys.144，923-947. (doi：10.1007/s10955-011-0285-9））型号。我们证明，在类比与代理为基础的模型，非本地对齐的存在下，强制执行强有力的解决方案，自组织成一个宏观的羊群。这就提出了存在这种强解的问题。我们解决这个问题，在一维和二维设置，证明全球次临界初始数据的规律性。事实上，我们表明，在相空间的初始配置，这决定了全球的正则性与有限时间爆破存在临界阈值。特别地，我们探讨了真空存在下的非局域排列的规律性。
We study the large-time behaviour of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e. g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52, 852-862. (doi:10.1109/TAC.2007.895842)) and Motsch & Tadmor (2011 J. Stat. Phys. 144, 923-947. (doi:10.1007/s10955-011-0285-9)) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one-and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.