We study the large-time behavior of a hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with the alignment which makes the large time behavior very different from the original Cucker–Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of quadratic potentials, we are able to treat a large class of admissible interaction kernels, $$\phi (r) > rsim (1+r^2)^{-\beta }$$ ϕ ( r ) ≳ ( 1 + r 2 ) - β with ‘thin’ tails $$\beta \leqslant 1$$ β ⩽ 1 —thinner than the usual ‘fat-tail’ kernels encountered in CS flocking $$\beta \leqslant \nicefrac {1}{2}$$ β ⩽ 1 2 ; we discover unconditional flocking with exponential convergence of velocities and positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a stability condition, requiring a large enough alignment kernel to avoid crowd scattering. We then prove, by hypocoercivity arguments, that both the velocities and positions of a smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.

中文翻译：

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具有外部电位的植绒流体动力学

我们研究了流体动力学模型的长时间行为，该模型描述了代理连续体的集体行为，由具有额外外部潜在强迫的成对对齐相互作用驱动。外力倾向于与对齐竞争，这使得大时间行为与原始的 Cucker-Smale (CS) 对齐模型非常不同，而且更加有趣。这里我们关注一致凸势。在二次势的特殊情况下，我们能够处理一大类可接受的相互作用核，$$\phi (r) > rsim (1+r^2)^{-\beta }$$ ϕ ( r ) ≳ ( 1 + r 2 ) - β 带有“细”尾 $$\beta \leqslant 1$$ β ⩽ 1 — 比 CS 植绒中遇到的通常“肥尾”核更薄 $$\beta \leqslant \nicefrac {1 {2}$$ β ⩽ 1 2 ; 我们发现了无条件聚集，速度和位置呈指数收敛，朝向作为谐振子运动的狄拉克质量。对于一般的凸势，我们强加了一个稳定性条件，需要一个足够大的对齐内核来避免人群分散。然后，我们通过矫顽力论证证明光滑解的速度和位置都必须聚集。我们还证明了一维和二维空间的全局平滑解的存在，受初始配置空间的临界阈值影响。有趣的是，可以保证次临界初始数据的全局平滑性，而与大时间聚集行为的先验知识无关。需要足够大的对齐内核以避免人群分散。然后，我们通过矫顽力论证证明光滑解的速度和位置都必须聚集。我们还证明了一维和二维空间的全局平滑解的存在，受初始配置空间中的临界阈值影响。有趣的是，可以保证次临界初始数据的全局平滑性，而与大时间聚集行为的先验知识无关。需要足够大的对齐内核以避免人群分散。然后，我们通过矫顽力论证证明光滑解的速度和位置都必须聚集。我们还证明了一维和二维空间的全局平滑解的存在，受初始配置空间的临界阈值影响。有趣的是，可以保证次临界初始数据的全局平滑性，而与大时间聚集行为的先验知识无关。