Orientation Mixing in Active Suspensions

IF 2.4 1区 数学 Q1 MATHEMATICS Annals of Pde Pub Date : 2023-10-20 DOI:10.1007/s40818-023-00163-8
Michele Coti Zelati, Helge Dietert, David Gérard-Varet
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引用次数: 6

Abstract

We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experience a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion \(\nu \) that the mixing estimates persist up to time scale \(\nu ^{-1/2}\) until the exponential decay at enhanced dissipation rate \(\nu ^{1/2}\) takes over. The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

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活性悬浮液中的定向混合
我们研究了Saintillan和Shelley为活性细长颗粒悬浮液动力学引入的一个流行的动力学模型,其中颗粒通过空间和方向的分布来描述。粒子的均匀分布是不相干的静止状态,已知它表现出相变。我们对非相干态的线性化演化进行了广泛的研究。我们表明(i)在与光谱(中性)稳定性相对应的非扩散状态下,悬架经历了类似于朗道阻尼的混合现象,并且我们在弱拓扑中提供了最佳的逐点时间衰减率。此外,我们证明了(ii)在小旋转扩散的情况下,混合估计一直持续到时间尺度上,直到以增强的耗散率(1/2)的指数衰减接管为止。有趣的特征是,动力学模型中通常的速度变量被球体上的方向变量所取代。相关的定向混合导致宏观量的有限代数衰减。为了证明,我们从Volterra方程的一般逐点衰减结果开始,该结果可能具有独立的兴趣。虽然在非扩散情况下,球体上的显式公式可以得出所需的衰变,但在扩散情况下需要做更多的工作:在这里,我们通过将优化的次高斯方法与矢量场方法相结合,证明了球体上平流-扩散方程的混合估计。本文中的一个要点是为球体上的平流-扩散算子确定良好的交换矢量场。我们在这个方向上的结果可能对涉及方向变量的集体动力学中的其他模型有用。
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来源期刊
Annals of Pde
Annals of Pde Mathematics-Geometry and Topology
CiteScore
3.70
自引率
3.60%
发文量
22
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