Abstract
This paper is concerned with the initial boundary value problem of the Vlasov–Fokker–Planck equation coupled with either the incompressible or compressible Navier–Stokes equations in a bounded domain. The global existence of unique strong solution and its exponential convergence rate to the equilibrium state are proved under the Maxwell boundary condition for the incompressible case and specular reflection boundary condition for the compressible case, respectively. For the compressible model, to overcome the lack of regularity due to the coupling with the kinetic equation in a bounded domain, an essential L10/3 estimate is analyzed so that the a priori estimate can be closed by applying the SPL theory developed by Guo et al. for kinetic models, [Arch Ration Mech Anal 236(3): 1389–1454 (2020)].
| Original language | English |
|---|---|
| Article number | 42 |
| Journal | Journal of Statistical Physics |
| Volume | 186 |
| Issue number | 3 |
| Online published | 5 Feb 2022 |
| DOIs | |
| Publication status | Published - Mar 2022 |
Funding
H.-L. Li’s research was supported by National Natural Science Foundation of China 11931010 and 11871047 and by the key research project of Academy for Multidisciplinary Studies, Capital Normal University, and by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds 007/20530290068. S.-Q. Liu’s research was supported by the National Natural Science Foundation of China 11971201 and 11731008. T. Yang’s research was supported by a fellowship award from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. SRF2021-1S01).
Research Keywords
- L2-L10/3 estimate
- Maxwell boundary condition
- Navier–Stokes–Vlasov–Pokker–Plack system
- Specular reflection boundary condition