Consider an open set $\Omega$ with smooth boundary and some smooth time-depending divergence-free vector field $u:[0,T]\times \Omega \rightarrow \mathbb{R}^d$, with $d=2$ or $3$, satisfying $u_{|\partial\Omega}=0$.
Intuitevily, because of the divergence-free condition and the vanishing at the boundary, I would guess that the vector field $u$ has to "loop" in some sense (it's even clearer in dimension $2$).
For $x\in \Omega$, I call $\mathscr{C}_x$ the trajectory issued from $x$, that is the solution of the Cauchy problem \begin{align*} Z'(t) &= u(t,Z(t)) \\ Z(0)&=x. \end{align*} My question : what can be said about the set of all $x\in \Omega$ such $\mathscr{C}_x$ is closed ? Is it always non-empty ? Can we prove that for every $a\in\Omega$, $a$ may be encircled by a trajectory of this kind ? What if I replace $u$ by $u \star \varphi$, where $\varphi$ is some smooth regularizing kernel ?
Thanks for any help !