# Global regularity for the fractional Euler alignment system

## TL;DR

This paper proves that a fractional Euler alignment system with nonlinear, density-dependent alignment and fractional dissipation has globally regular solutions, highlighting a novel non-local nonlinear regularization mechanism.

## Contribution

It demonstrates for the first time that nonlinear, density-dependent alignment enhances dissipation, ensuring global regularity in a fractional Euler system.

## Key findings

- Solutions are globally regular for all fractional orders in (0, 1)
- Non-local nonlinear modulation of dissipation prevents shock formation
- Alignment term enhances dissipation leading to regularity

## Abstract

We study a pressureless Euler system with a nonlinear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian $(-\partial_{xx})^{\alpha/2}$, $\alpha \in (0, 1)$. The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all $\alpha \in (0, 1)$. To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.