# Steady Euler flows with compact support?

What is known about (3D) steady incompressible Euler flows with compact support?

(Looking for results in a field you are not familiar with sure is tough. I had a hope to find clues starting from the famous paper of V. Scheﬀer about a flow with compact support in space-time, and from works on vortex rings I could find, but in both cases ended up empty handed. This problem was considered by some, I presume? )

According to the following paper, it is an open problem whether such solutions exist: N. Nadirashvili, Liouville theorem for Beltrami flow, Geometric and Functional Analysis 24 (2014), 916-921.

• Thank you. I have two problems with this source though: 1. There is no reference at all. 2. The space is not mentioned (no solution even in $L^2$?) – Alex Gavrilov Mar 1 '18 at 5:22
• @AlexGavrilov: How would you give a reference to substantiate that a problem is open? I mean, other than by citing someone else who has made the same statement. – Michael Renardy Mar 1 '18 at 18:04
• I simply suspect that the problem is open but very obscure and not interesting to anyone besides a couple of experts. Actually, I had plans to try and construct an example, but given the response I am about to change my mind. – Alex Gavrilov Mar 2 '18 at 12:00