Vlasov equation: weak and long-range vs weak or long-range? Recently I asked this question on why we need long-range interactions for the Vlasov equation – something I now understand.  I have also seen several derivations* explain individually why we need long-range vs weak interactions.  From what I can tell these assumptions seem to do the same thing and I have yet to see a resource explain why we actually need them both.
So my question is:  For the Vlasov equation to be valid are we required to have an interaction which is both long-range and weak or can we get away with just long-range or just weak?
* I don't have access to the resources these are in now but will update the question when I can get hold of them.
 A: Consider the lowest-order entry of the BBGKY hierarchy that governs the evolution of the one-particle distribution function, $f_1(r, v, t)$, in terms of the two-particle distribution function, $f_2(r, r', v, v', t)$:
$$
\frac{\partial f_1}{\partial t} + v\cdot\nabla_r f_1 + \frac{F(r,t)}{m}\cdot\nabla_v f_1
= \int d^3r'd^3v' \frac{1}{m}\nabla_rV(r-r')\cdot\nabla_v f_2
$$
Here, $F(r,t)$ is an external force, and $V(r-r')$ is the interaction potential between particles. Now decompose $f_2$ into two parts: 
$$ f_2(r,r',v,v',t) = f_1(r,v,t)f_1(r',v',t) + g_2(r,r',v,v',t)$$
The first term represents the statistically uncorrelated part of $f_2$, and the second term represents the correlations between particles due to close interactions ("collisions"). For weak interactions, one may neglect $g_2$ and write
$f_2(r,r',v,v',t)\approx f_1(r,v,t)f_1(r',v',t)$. Substituting into the BBGKY equation gives
$$
\frac{\partial f_1}{\partial t} + v\cdot\nabla_r f_1 + \frac{F(r,t)}{m}\cdot\nabla_v f_1
= \frac{1}{m}\left[\int d^3r'd^3v' f_1(r',v',t)\nabla_r V(r-r')\right]\cdot\nabla_v f_1(r,v,t) 
$$
Observe that the quantity in square brackets is minus the mean force of interaction between two particles - the "Vlasov mean field," $\overline{F}(r,t)$. If we bring this term over to the left-hand side of the equation, we arrive at the "Vlasov equation"
$$
\frac{\partial f_1}{\partial t} + v\cdot\nabla f_1 + \frac{1}{m}[F(r,t)+\overline{F}(r,t)]\cdot\nabla_v f_1
= 0
\\
\overline{F}(r,t) = -\nabla_r\int d^3r'd^3v' V(r-r')f_1(r',v',t)
$$
In the context of plasma, $\overline{F}(r,t)$, is the force arising from the electric field of the charges themselves. 
Let us now wonder: why did we neglect $g_2$ but not $\overline{F}$? They are both associated with the pairwise interactions between particles; shouldn't they  both be negligible in the "collisionless" limit? More precisely, under what conditions should we neglect
$$
\int d^3r'd^3v' \frac{1}{m}\nabla_r V(r-r')\cdot\nabla_v g_2(r,r',v,v',t)
$$
but not the mean field? The answer lies in the range of the interaction $V$. 
Suppose $V$ is short-ranged, effectively vanishing for $|r-r'|>\sigma$. In the context of neutral gases, $\sigma$ would represent the effective diameter of the particles. It stands to reason that $g_2$ will also vanish beyond this range. In this case, both the neglected collision term and the mean-field term are of the same order, since they both involve integrals over $r'$ that vanish except when the two particles are within a distance $\sigma$ from one another. 
However, what if $V$ is the Coulomb interaction, whose range in infinite? In this case, the mean field integral over $r'$ must extend over all space. On the other hand, $g_2$ will still have a finite extent, since Debye screening will cause pair correlations to die out when $|r-r'| > \lambda_D$, the Debye length. For this reason, the "collision" term with $g_2$ is of smaller order than the mean-field term.
So to answer your question directly now: within the approximation of collisionlessness, the Vlasov equation is valid regardless of the range of the interaction potential. However, the mean-field term must be retained if the interaction is long-ranged and must be dropped if it is short-ranged.
side note: The mean-field term is really the hallmark of the Vlasov equation. It is responsible for many of the interesting kinetic phenomena in plasma that are not present in neutral gases, e.g., Landau damping and Debye screening. 
