Solutions at vacuum and rarefaction waves in pressureless Euler alignment system

TL;DR

This paper constructs global weak solutions for the pressureless Euler alignment system on the real line, allowing vacuum states, and analyzes their asymptotic behavior showing density spreading and velocity approaching a rarefaction wave.

Contribution

It provides a method to construct global weak solutions with vacuum and non-smoothness, and studies their long-term asymptotic behavior under rescaling.

Findings

Solutions exhibit vacuum states and non-smoothness.

Density becomes uniformly distributed over expanding intervals.

Velocity converges to a rarefaction wave.

Abstract

We construct global-in-time weak solutions to the pressureless Euler alignment system posed on the whole line and supplemented with initial conditions, where an initial density is an arbitrary, nonnegative, bounded, and integrable function (hence density at vacuum is allowed) and the corresponding initial velocity is determined by certain inequalities. Moreover, our setting covers the case where solutions to the pressureless Euler alignment system are known to be non-smooth. We also study an asymptotic behavior of constructed solutions and we show that, under a suitable rescaling, the density looks like a uniform distribution on a bounded, time dependent, expanding-in-time interval and the corresponding velocity approaches a rarefaction wave (i.e. the well-known explicit solution to the inviscid Burgers equation).