Another recent paper that deals with this question is this one, where they get "as much regularity as possible":
http://cvgmt.sns.it/paper/3700/
The spirit is Eulerian and they use the convex integration technique (so unluckily no explicit vector field): they show that basically they can reach whatever smooth measure you want (besides the lebesgue measure), with a sobolev vector field.
However it is set on the torus and so you don't have the compact support hypotesis; but the same technique has been used also with the euler equation, and in that case they could also have a counterexample to uniqueness with compact support, so I would expect this is also the case.
Hope this helps!