Abstract
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 Schrödinger-type equations in quantum mechanics.
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