Some remarks on steady solutions to the Euler system in $\mathbb{R}^d$

TL;DR

This paper demonstrates the existence of infinitely many steady solutions with compact velocity support for the Euler system in two and three dimensions, including incompressible cases with variable density, highlighting complex solution structures.

Contribution

It proves the existence of infinitely many steady solutions with compact support for Euler systems in multiple dimensions, extending understanding of solution multiplicity in fluid dynamics.

Findings

Existence of infinitely many steady solutions with compact velocity support.

Results apply to both gas dynamics and incompressible Euler systems with variable density.

These solutions are globally smooth and non-trivial.

Abstract

We show that the Euler system of gas dynamics in $\mathbb{R}^d$, $d=2,3$, with positive far field density and arbitrary far field entropy, admits infinitely many steady solutions with compactly supported velocity. The same proof yields a similar result for the incompressible Euler system with variable density. In particular, these are examples of global in time smooth (non-trivial) solutions for the corresponding time-dependent systems.