Alignment with nonlinear velocity couplings: collision-avoidance and micro-to-macro mean-field limits

Abstract: We investigate the pressureless fractional Euler-alignment system with nonlinear velocity couplings, referred to as the $p$-Euler-alignment system. This model features a nonlinear velocity alignment force, interpreted as a density-weighted fractional $p$-Laplacian when the singularity parameter $\alpha$ exceeds the spatial dimension $d$. Our primary goal is to establish the existence of solutions for strongly singular interactions ($\alpha \ge d$) and compactly supported initial conditions. We construct solutions as mean-field limits of empirical measures from a kinetic variant of the $p$-Euler-alignment system. Specifically, we show that a sequence of empirical measures converges to a finite Radon measure, whose local density and velocity satisfy the $p$-Euler-alignment system. Our results are the first to prove the existence of solutions to this system in multi-dimensional settings without significant initial data restrictions, covering both nonlinear ($p>2$) and linear ($p=2$) cases. Additionally, we establish global existence, uniqueness, and collision avoidance for the corresponding particle ODE system under non-collisional initial conditions, extending previous results for $1 \le p \le \alpha + 2$. This analysis supports our mean-field limit argument and contributes to understanding alignment models with singular communication.