Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations

We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vlasov-Navier-Stokes equations. When the initial data is sufficiently small and regular, we first show the unique existence of the global strong solution to the kinetic-fluid equations, and establish the a priori estimates for the large-time behavior using an appropriate Lyapunov functional. More specifically, we show that the velocities of particles and fluid tend to be aligned together exponentially fast, provided that the local density of the particles satisfies a certain integrability condition.