Large-time asymptotics of periodic two-dimensional Vlasov-Navier-Stokes flows

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [17] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the non-homogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.