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∞-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∈[1,∞], 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.
|Journal||Nonlinear Analysis: Real World Applications|
|Publication status||Published - 2022 Feb|
Bibliographical noteFunding Information:
Y.-P. Choi’s work is supported by NRF, Republic of Korea - 2017R1C1B2012918 , POSCO Science Fellowship of POSCO TJ Park Foundation, Republic of Korea , and Yonsei University, Republic of Korea Research Fund of 2019-22-0212 . K. Kang’s work is partially supported by NRF - 2019R1A2C1084685 and NRF - 2015R1A5A1009350 . H. K. Kim’s work is supported by NRF - 2018R1D1A1B07049357 . J.-M. Kim’s work is supported by NRF - 2020R1C1C1A01006521 and a Research Grant of Andong National University .
Y.-P. Choi's work is supported by NRF, Republic of Korea-2017R1C1B2012918, POSCO Science Fellowship of POSCO TJ Park Foundation, Republic of Korea, and Yonsei University, Republic of Korea Research Fund of 2019-22-0212. K. Kang's work is partially supported by NRF-2019R1A2C1084685 and NRF -2015R1A5A1009350. H. K. Kim's work is supported by NRF -2018R1D1A1B07049357. J.-M. Kim's work is supported by NRF -2020R1C1C1A01006521 and a Research Grant of Andong National University.
© 2021 Elsevier Ltd
All Science Journal Classification (ASJC) codes
- Economics, Econometrics and Finance(all)
- Computational Mathematics
- Applied Mathematics