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.

我们关注二维 Vlasov-Navier-Stokes 方程和三维 Vlasov-Stokes 系统的解的大时间行为，包括速度对齐/未对齐的影响。我们首先重新审视我们主系统的长时间行为估计，并改进对维度和通信权重函数的假设。特别是，这使我们能够考虑粒子之间未对准相互作用的影响。然后我们使用一个尖锐的热核估计来获得流体速度的指数时间衰减到它的平均值${升}^{\infty }$-规范。对于动力学部分，通过采用由平均粒子速度调制加权的某种类型的 Sobolev 范数，我们证明了粒子分布的指数时间衰减，前提是局部粒子分布函数是均匀有界的。此外，我们表明，粒子分布函数对速度的支持收缩到一个点，即平均初始粒子和流体速度的平均值，随着时间的推移呈指数增长。这也规定，对于任何$磷\in [1,\infty ]$， 这 $磷$-Wasserstein 距离粒子分布函数与局部粒子分布的张量积和速度在该点的狄拉克测度之间的距离随着时间趋于无穷大以指数方式快速收敛到零。