# Global well-posedness for the Euler alignment system with mildly singular interactions

## TL;DR

This paper proves the global regularity of solutions for the Euler alignment system with less singular interaction kernels, extending previous results to a broader class of kernels that are not integrable, using modulus of continuity techniques.

## Contribution

It establishes global well-posedness for the Euler alignment system with mildly singular, non-integrable interaction kernels, broadening the class of kernels for which regularity is known.

## Key findings

- Global regularity for less singular kernels established
- Solutions remain smooth as long as kernels are non-integrable
- Method relies on modulus of continuity estimates for integro-differential equations

## Abstract

We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved in\cite{do2018global,shvydkoy2017eulerian1,shvydkoy2017eulerian2,shvydkoy2017eulerian3}. Here, we consider a class of less singular interaction kernels and establish the global regularity of solutions as long as the interaction kernels are not integrable. The proof relies on modulus of continuity estimates for a class of parabolic integro-differential equations with a drift and mildly singular kernels.