Soft question: What does it mean if the area integral of the divergence of a vector field over a region is negative? Let $\vec{F}(x,y)$ be a vector field over $\mathbb{R}^2$ and let $\nabla \cdot \vec{F}$ be its divergence.  Presume that $\delta$ is a circle in the $x,y$ plane.  If I compute:
$$I = \int\int_\delta \nabla \cdot \vec{F}dA$$
and find that $I < 0$, what does that mean?  Is it correct to say that the net flux of the vector field acrosss the boundary of that region is negative?  Is it further correct to say that $\vec{F}$ has more sinks than sources in that region?
 A: The answers to your two questions in the post are "yes" and "yes". As for your question in the comments about the velocity field, the answer is no. As a concrete example, consider $F:\Bbb{R}^2\to\Bbb{R}^2$ defined as $F(x,y)=(x,-2y)$. Then, $\nabla \cdot F = -1$, so it has constant negative divergence (by slight modification, this works in any number of dimensions).
On the other hand, the integral curves with initial condition at $p=(a,b)\in\Bbb{R}^2$ are given by
\begin{align}
\gamma_p(t)&= (ae^t, be^{-2t})
\end{align}
In other words, if you start your particle at position $p=(a,b)$, and you let it "flow along the vector field" then the above curves are what you get (you can easily see they satisfy $\gamma_p'(t)=F(\gamma_p(t))$ together with the initial condition $\gamma_p(0)=p$).
Look closely at the solution, it has the $e^t$ term which explodes. So, as a very concrete illustration, suppose $p=(1,0)$, so your particle is originally 1 unit away on x axis. Them, $\gamma_{(1,0)}(t)=(e^t,0)$, so as time goes on the particle goes off to infinity, even though the vector field has constant negative divergence.
This apparent mismatch comes from the fact that the integral $\int_D\text{div}(F)\,dA = \int_{\partial D}F\cdot n\, dl$ measures the flux across the entire boundary, which is like a "net effect". On the other hand, we can focus our attention to specific directions (like the $x$-axis in my example), where everything could be exploding, and totally disregard the other directions.
