3 The characteristic flow and well-posedness
The Vlasov equation transports mass along the characteristic system \(\dot X = V\), \(\dot V = -(\nabla W * \rho _t)(X)\). Existence is a fixed-point argument in the space of measure curves: freeze the field, solve the frozen linear problem by transporting the datum along its flow, and contract in the \(W_1\)-sup metric on a short window. Windows of a fixed length depending only on \(L\) then glue to arbitrary horizons, so no smallness of the interaction is required — Theorem 1.3 holds at every Lipschitz constant.
The phase-space velocity field \(b(t,x,v) = \big(v,\; -(\nabla W * \rho _t)(x)\big)\) generating the characteristic flow. Under Assumption 1 it is Lipschitz in the phase variable, with constant governed by \(\max (1, L)\).
The window-localized weak-solution predicate: the distributional Vlasov equation holds on the open interval \((0,T)\). The predicate carries only the PDE; continuity in time, where needed, is a separate hypothesis (as in Theorem 1.7 of the paper). Local existence lives on windows; the global theorem glues them.
The window-localized Lagrangian-solution predicate on \([0,T]\): a weak solution on the window together with a characteristic-flow witness transporting the initial datum.
Let \(W\) satisfy Assumption 1 and let \(f_0\) be a probability measure with finite first moment. The forward Cauchy problem is well-posed: there exists a single curve \(f : [0,\infty ) \to \mathcal{P}_1(\mathbb {R}^d \times \mathbb {R}^d)\) with datum \(f(0) = f_0\) and finite first moment at every \(t \ge 0\), which is a Lagrangian solution on every window \([0,T]\); and on each window it is the unique Lagrangian solution with that datum.
Freeze the field and run Picard iteration on curves of measures: on a short window the solution map is a contraction in the \(W_1\)-sup metric, with ratio controlled by \(L \cdot T\), so Banach’s fixed-point theorem yields a unique local Lagrangian solution. Windows of a fixed length depending only on \(L\) are then glued to reach arbitrary horizons — no smallness of \(L\) is required. Per-window uniqueness follows from the stability estimate of the next chapter for \(L {\gt} 0\), and is explicit in the degenerate constant-force case \(L = 0\).