Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

TL;DR

This paper establishes global existence, uniqueness, and decay properties of solutions to a compressible Euler system with velocity alignment in critical spaces, advancing understanding of its long-term behavior.

Contribution

It proves global well-posedness and asymptotic behavior for the compressible Euler system with singular velocity alignment in critical Besov spaces, which is a novel result.

Findings

Existence and uniqueness of global solutions for small initial data.

Large-time asymptotic behavior and decay estimates of solutions.

Local-in-time solvability also addressed.

Abstract

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behavior and optimal decay estimates of the solutions as $t\to \infty$.