Large-time behavior of small-data solutions tothe Vlasov–Navier–Stokes system on the whole space
Daniel Han-Kwan
Abstract
We study the large time behavior of small data solutions to the\nVlasov-Navier-Stokes system on $\\R^3 \\times \\R^3$. We prove that the kinetic\ndistribution function concentrates in velocity to a Dirac mass supported at\n$0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate.\nThe proof is based on two steps, following the general strategy laid out in\n\\cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming\na uniform control of the kinetic density, (2) a bootstrap argument allows to\nobtain such a control. This last step requires a fine understanding of the\nstructure of the so-called Brinkman force, which follows from a family of new\nidentities for the dissipation (and higher versions of it) associated to the\nVlasov-Navier-Stokes system.\n