ASYMPTOTIC BEHAVIOR OF THE NONLINEAR VLASOV EQUATION WITH A SELF-CONSISTENT FORCE

We present a critical threshold phenomenon on the L-1-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 L-1-asymptotic completeness, which means that the nonlinear Vlasov flow cannot be approximated by the corresponding free flow in L-1-norm time-asymptotically. 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 Schrodinger-type equations in quantum mechanics.