a way to assign a vector field $v_t$ to the distribution $d\mu_t/dt$, and this vector field gives a notion of “direction of the movement.” The vector field $v_t$ is not unique: it is ambiguous up to addition of a vector field $u_t$ such that $\operatorname{div}(\mu_tu_t)=0$ for all $t\in{\mathbb{R}}$. Since the set of possible $u_t$ is a closed subspace of $L^p(\mu_t)$, one can choose the vector field $v_t$ to be the minimizer of the $L^p(\mu_t)$ norm for each $t$. It has been shown [@ambrosiogiglisavare] that the minimizer is in fact a gradient vector field, $v_t=\nabla\phi_t$ for some functions $\phi_t\colon{\mathbb{R}}^n\to{\mathbb{R}}$.

In the theory of $W_p$-absolutely continuous families of measures, $v_t$ exists only for almost every $t$. Consider the situation of Example \[ex:explicit\]: the family $\mu_t=\rho(x,t)dx$ given in the example is $W_p$-absolutely continuous, but $v_0$ is not defined for $k{\geqslant}1$. The results of Section \[sec:characterization\] indicate that in cases like this, and surely also in the cases of families of measures that are not $W_p$-absolutely continuous but are still weakly differentiable, the distributions that can arise as their derivatives can be much more general than those of the form $\operatorname{div}(\mu_tv_t)$, for some vector field $v_t$; cf. Example \[ex:uniform\]. As it turns out, an 