I am looking for a (possibly time-dependent) vector field $v: [0,1] \times \mathbb R_x^d \to \mathbb R^d$ such that

1. $\text{div}_x v = 0$ ;

2. $v$ 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 $i=1,2$;

3. $v$ 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.

I am looking for a (possibly time-dependent) vector field $v: [0,1] \times \mathbb R_x^d \to \mathbb R^d$ such that

1. $\text{div}_x v = 0$ ;

2. $v$ 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 $i=1,2$;

3. $v$ 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.

I am looking for a vector field $v: [0,1] \times \mathbb R_x^d \to \mathbb R^d$ such that

1. $\text{div}_x v = 0$ ;

2. $v$ 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 $i=1,2$;

3. $v$ 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.

I am looking for a (possibly time-dependent) vector field $v: [0,1] \times \mathbb R_x^d \to \mathbb R^d$ such that

1. $\text{div}_x v = 0$ ;

2. $v$ 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 $i=1,2$;

3. $v$ 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.