论文标题

vlasov方程的时间衰减和渐近行为,其植入术语与不可压缩的流体流相结合

Temporal decays and asymptotic behaviors for a Vlasov equation with a flocking term coupled to incompressible fluid flow

论文作者

Choi, Young-Pil, Kang, Kyungkeun, Kim, Hwa Kil, Kim, Jae-Myoung

论文摘要

我们关注的是,在二维和弗拉索夫 - 斯托克斯系统中,弗拉索夫的解决方案的大型解决方案在三个方面,包括速度比对/错位的效果。我们首先重新审视主系统的大型行为估计,并完善对维度和通信权重函数的假设。特别是,这使我们能够考虑到粒子之间未对准相互作用的影响。然后,我们使用尖锐的热核估计值将流体速度的指数时间衰减达到其平均水平,以$ l^\ infty $ norm的平均值。对于动力学部分,通过采用通过平均粒子速度调制的某种类型的Sobolev规范,我们证明了粒子分布的指数时间衰减,只要局部粒子分布均匀地界限。此外,我们表明,速度中粒子分布函数的支持会缩小到一个点,这是平均初始粒子和流体速度的平均值,随着时间的流逝,它呈指数速度。这还规定,对于[1,\ infty] $中的任何$ p \,粒子分布函数与局部粒子分布的张量产品与当时dirac量的张量产品之间的$ p $ - wasserstein距离,速度的速度量,随着时间的流逝,速度的速度将速度呈成倍地转化至零。

We are concerned with large-time behaviors of solutions for Vlasov--Navier--Stokes equations in two dimensions and Vlasov-Stokes system in three dimensions including the effect of velocity alignment/misalignment. We first revisit the large-time behavior estimate for our main system and refine assumptions on the dimensions and a communication weight function. In particular, this allows us to take into account the effect of the misalignment interactions between particles. We then use a sharp heat kernel estimate to obtain the exponential time decay of fluid velocity to its average in $L^\infty$-norm. For the kinetic part, by employing a certain type of Sobolev norm weighted by modulations of averaged particle velocity, we prove the exponential time decay of the particle distribution, provided that local particle distribution function is uniformly bounded. Moreover, we show that the support of particle distribution function in velocity shrinks to a point, which is the mean of averaged initial particle and fluid velocities, exponentially fast as time goes to infinity. This also provides that for any $p \in [1,\infty]$, the $p$-Wasserstein distance between the particle distribution function and the tensor product of the local particle distributions and Dirac measure at that point in velocity converges exponentially fast to zero as time goes to infinity.

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