On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity

A well-known result of Carrillo, Choi, Tadmor, and Tan states that the 1D Euler Alignment model with smooth interaction kernels possesses a 'critical threshold' criterion for the global existence or finite-time blowup of solutions, depending on the global nonnegativity (or lack thereof) of the quantity $e_0 = \partial_x u_0 + \phi*\rho_0$. In this note, we rewrite the 1D Euler Alignment model as a first-order system for the particle trajectories in terms of a certain primitive $\psi_0$ of $e_0$; using the resulting structure, we give a complete characterization of global-in-time existence versus finite-time blowup of regular solutions that does not require a velocity to be defined in the vacuum. We also prove certain upper and lower bounds on the separation of particle trajectories, valid for smooth and weakly singular kernels, and we use them to weaken the hypotheses of Tan sufficient for the global-in-time existence of a solution in the weakly singular case, when the order of the singularity lies in the range $s\in (0,\frac12)$.