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

TL;DR

This paper introduces a new technique to analyze the Euler-alignment system with weakly singular kernels, enabling global existence results under weaker conditions by bounding the influence function convolution more effectively.

Contribution

It presents a novel approach to bounding the influence function convolution, relaxing previous assumptions and extending the analysis to a broader class of weakly singular kernels.

Findings

Established global-in-time existence results.

Bounded the influence function convolution by a relaxed constant.

Provided refined solution bounds based on kernel structure.

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 $\psi$.