We establish the large-time behavior for the coupled kinetic-fluid equations. More precisely, we consider the Vlasov equation coupled to the compressible isentropic Navier-Stokes equations through a drag forcing term. For this system, the large-time behavior shows the exponential alignment between particles and fluid velocities as time evolves. This improves the previous result by Bae et al. [Discrete Contin. Dyn. Syst. 34, 4419-4458 (2014)] in which they considered the Vlasov/Navier-Stokes equations with nonlocal velocity alignment forces for particles. Employing a new Lyapunov functional measuring the fluctuations of momentum and mass from the averaged quantities, we refine assumptions for the large-time behavior of the solutions to that system.
Bibliographical noteFunding Information:
The author warmly thanks the anonymous referee for helpful comments. The author was supported by Engineering and Physical Sciences Research Council (Grant No. EP/K008404/1) and the ERC-Starting Grant HDSPCONTR "High-Dimensional Sparse Optimal Control." The work is supported by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics