Regular solutions to the fractional Euler alignment system in the Besov spaces framework

TL;DR

This paper constructs both local and global regular solutions for the fractional Euler alignment system in the whole space, using Besov spaces to handle the density's transport equation without damping.

Contribution

It introduces a novel functional framework based on Besov spaces for analyzing the fractional Euler alignment system, enabling new regularity results.

Findings

Existence of large local solutions

Existence of small global solutions

Use of Besov spaces for density estimates

Abstract

We here construct (large) local and small global-in-time regular unique solutions to the fractional Euler alignment system in the whole space ${\mathbb R}^d$, in the case where the deviation of the initial density from a constant is sufficiently small. Our analysis strongly relies on the use of Besov spaces of the type $L^1(0,T;\dot B^s_{p,1})$, which allow to get time independent estimates for the density even though it satisfies a transport equation with no damping. Our choice of a functional setting is not optimal but aims at providing a transparent and accessible argumentation.