Non-uniqueness of flow for divergence free vector fields 
I am looking for a (possibly time-dependent) vector field $v: [0,1] \times \mathbb R_x^d \to
 \mathbb R^d$ such that 
  
  
*
  
*$\text{div}_x  v = 0$ ; 
  
*$v$ has more than one (measure-preserving) flow, i.e. there exist two different maps $X_1,X_2$ such that 
  $$X_i(t,x) = x + \int_0^t v(s,X_i(s,x))\,ds$$ for
  every $x \in \mathbb R^d$ and $t \in [0,1]$, for $i=1,2$;
  
*$v$ is compactly supported (say in the unit ball) and bounded.

The paper by Aizenman, M. On Vector Fields as Generators of Flows: A Counterexample to Nelson's Conjecture (Annals of Mathematics, Second Series, 107, no. 2 (1978): 287-96. doi:10.2307/1971145) seems to address the points 1. & 2. but I have no idea if one can include also point 3. I believe that, given a vector field solving 1 & 2, it should be rather easy to cook up one satisfying 1,2 & 3 but I do not see how to handle it. There are also some related results in the periodic setting by Depauw. 
 A: The problems seems far from easy. This very recent paper gives an example:
https://arxiv.org/pdf/1611.05928.pdf .
A: On $\mathbb{R}^2$, what about : 
$$v(x,y)=
\begin{cases} e_y  & \text{ if } y\geq |x| \\ 
-e_y & \text{ if } -y\geq |x| \\ 
-e_x & \text{ if } x\geq |y| \\ 
e_x  & \text{ if } -x\geq |y| \end{cases}  \\ $$ ($e_x,e_y$: the canonical basis of $\mathbb{R}^2$)?
A: Another recent paper that deals with this question is this one:
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!
