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

TL;DR

This paper provides a new characterization of the global existence and blowup of solutions in the 1D Euler Alignment model without vacuum velocity, using particle trajectories and primitive functions, extending previous criteria.

Contribution

It rewrites the model as a first-order system for trajectories, removing the need for velocity in vacuum, and weakens conditions for global existence in weakly singular cases.

Findings

Complete characterization of solution behavior without vacuum velocity.

Bounds on particle trajectory separation for smooth and weakly singular kernels.

Weakened hypotheses for global existence in the weakly singular case.

Abstract

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 $s\in (0,\frac12)$.