A Necessary and Sufficient Condition for the Existence of Global Smooth Solutions to the One-Dimensional Inviscid Burgers-Vlasov System
We investigate global smooth solvability for the one-dimensional inviscid Burgers-Vlasov system. For a smooth initial fluid velocity and a nonnegative, compactly supported particle distribution, we establish a sharp necessary and sufficient threshold expressed through the sum of the initial velocity gradient and the initial particle density. The principal difficulty is to control simultaneously the nonlinear steepening of the fluid velocity, the drag coupling, the particle velocity support, and the full hierarchy of derivatives. To overcome it, we introduce a combined gradient-density quantity whose evolution closes along fluid characteristics. Its sign invariance, together with characteristic support estimates and fixed-order derivative bounds, yields global continuation whenever the initial threshold is nonnegative. If the threshold fails at one point, a Riccati comparison gives an explicit finite upper bound for the maximal smooth lifespan and identifies negative velocity-gradient blow-up when this bound is attained. We also use nonnegative regularization and compactness to extend the threshold result to initial data of any fixed finite regularity.