On the global smooth solution to 2-D fluid/particle system

In two space dimension, we show that the steady state the solution of fluid/particle system may tend to after a long time is completely determined by the initial total momentum. Based on this observation, we prove the global-in-time existence of the classical solutions for arbitrary initial data to the system that couples the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. By linearized method and Littlewood-Paley analysis, the exponential rate of the convergence toward steady state is obtained under some specific assumptions.