Global dynamics of the one-dimensional Euler-alignment system with weakly singular kernel

Abstract

This letter studies the Euler-alignment system with weakly singular influence functions by introducing a novel technique to bound the density. Instead of resorting to a nonlinear maximum principle used in Tan (2020) to bound the interaction term ψ∗ρ by ρs with s∈(0,1) for ψ with an algebraic singularity at origin, we bound ψ∗ρ by a relaxed constant for any ψ∈L1. We thus establish the global-in-time existence results with weaker assumptions on ψ and refined solution bounds, characterized by the structure of ψ.