Suppose we have a differentiable vector field $X:\Omega\to\mathbb{R^n}$ defined on an open, bounded and simply connected region subset $\Omega$ of $\mathbb{R^n}$, and its divergence is negative everywhere, i.e. $\nabla\cdot X(x)<0$ for any $x\in\Omega$. Can we prove that any two solution trajectories will evolve closer and closer, and thus eventually convergent?
Formally, given any two points $p,q\in\Omega$ and flows $p(t), q(t)$ satisfying $p(0)=p,q(0)=q,\dot p(t)=X(p(t)), \dot q(t)=X(q(t))$, define $f(t)=||p(t)-q(t)||^2$, then is it true that $$\frac{d}{dt}f(t)<0$$
Intuitively, I think it is right because negative divergence implies any closed area will evolve smaller and smaller. So if we enclose two points with a thin tube, then the volume of this tube will get smaller and smaller, which forces points get closer.
Here is my try: $$\frac{d}{dt}f(t)=2\left<p-q|X(p)-X(q)\right>=2\left<p-q|\nabla X(r)|p-q\right>$$ The first equation follows from the definition and the second from mean value theorem(though this theorem doesn't exist). But this seems the statement requires more rigid condition, say positiveness of $\nabla X$, to guarantee the corectness.
Let $g$ be a smooth, non-negative function (not identically zero) with support contained in some interval $[0, R^{2}]$. The vector field $$ X(x, y, z) = g(x^{2} + y^{2})(-y, x, 0) + (0, 0, kz) $$ defined on the open cylinder $$ \Omega = \{(x, y, z) : x^{2} + y^{2} < R^{2},\ |z| < 1\}, $$ is complete (the flow exists for all positive time) and has divergence $-k$, but a pair of orbits do not approach each other pointwise in any reasonable sense. Indeed, the flow can be written explicitly: If $r^{2} = x^{2} + y^{2}$, then $$ \Phi_{t}(x, y, z) = \left(x\cos\bigl(g(r^{2})t\bigr) - y\sin\bigl(g(r^{2})t\bigr), y\cos\bigl(g(r^{2})t\bigr) + x\sin\bigl(g(r^{2})t\bigr), ze^{-kt}\right). $$ Geometrically, a point $(x, y, z)$ at distance $r$ from the $z$-axis travels in a circle around the $z$-axis by a distance of $rg(r^{2})$ per unit time (and the aggregate motion is incompressible in the "horizontal" directions) while the third component decays exponentially to $0$.