Asymptotic behavior of the nonlinear vlasov equation with a self-consistent force

We present a critical threshold phenomenon on the L¹-asymptotic completeness for the nonlinear Vlasov equation with a self-consistent force. For a long-ranged self-consistent force, we show that the nonlinear Vlasov equation has no L¹-asymptotic completeness, which means that the nonlinear Vlasov flow cannot be approximated by the corresponding free flow in L ¹-norm timeasymptotically. In contrast, for a short-ranged force, the nonlinear Vlasov flow can be approximated by the free flow time-asymptotically. Our result corresponds to the kinetic analogue of scattering results to the Schr̈odinger-type equations in quantum mechanics.