Global dynamics of the 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 [C. Tan, Nonlinearity, 33: 1907--1924, 2020] to bound the interaction term $\psi\ast\rho$ by $\rho^s$ with $s\in (0, 1)$ for $\psi$ with an algebraic singularity at origin, we bound $\psi*\rho$ by a relaxed constant for any $\psi \in L^1$. We thus establish the global-in-time existence results with weaker assumptions on $\psi$ and refined solution bounds, characterized by the structure of $\psi$.