Global Regularity for the Fractional Euler Alignment System
Global Regularity for the Fractional Euler Alignment System
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DOI：
10.1007/s00205-017-1184-2
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发表时间：
2018-04-01
影响因子：
2.5
通讯作者：
Tan, Changhui
中科院分区：
文献类型：
--
作者：
Do, Tam;Kiselev, Alexander;Tan, Changhui
本文研究了一个起源于Cucker-Smale群集模型的具有非线性密度依赖排列项的无压Euler系统。在趋向于平衡速度的意义上，对准项是耗散的。它的密度依赖性是自然的：由于物种的不适，在高密度区域的对齐率增加。扩散项具有分数拉普拉斯算子的阶。相应的Burgers方程与线性耗散的这种类型的发展冲击在有限的时间。我们发现，对齐的非线性增强耗散，和解决方案是全局正则的。据我们所知，这是由于非局部非线性耗散调制的正则化的第一个例子。
We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian . The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all . To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.