the subset of $\mathcal G({\mathbb{R}}^n)$ associated to $\mu$. Note that $[\mu]$ always contains representatives $(f_\varepsilon)_\varepsilon\in[\mu]$ satisfying $$\label{eq:regularcolombeau}
\int_{{\mathbb{R}}^n}f_\varepsilon(x)\,dx=1\quad\textrm{and}\quad f_\varepsilon{\geqslant}0\quad \textrm{for $0<\varepsilon<1$}$$ because these can be obtained by convolution of $\mu$ with a mollifier of mass 1. Let $\mathcal V(\mu)$ be the set of families $(v^\varepsilon)_{0<\varepsilon<1}$ of smooth vector fields such that, for all representatives satisfying , we have $$(\operatorname{div}(f_\varepsilon v^\varepsilon))_{0<\varepsilon<1}\in \mathcal E({\mathbb{R}}^n).$$ Note that the set $\mathcal V(\mu)$ is a vector space.

Choose a representative $(g_\varepsilon)_{0<\varepsilon<1}$ in the subset $[\eta]$ that additionally satisfies that the support of $g_\varepsilon$ is contained in the support of $f_\varepsilon$; by Remark \[rmk:support\], such a representative can always be constructed by convolution.

Consider the continuity equation $$\label{eq:continuitysmooth}
 g_\varepsilon+\operatorname{div}(f_\varepsilon v^\varepsilon)=0.$$ The space $\mathcal V(\mu)$ always contains a solution $(v^\varepsilon)_{0<\varepsilon<1}$ to this equation; we know this because for each $\varepsilon$ this is just the classical smooth case, treated in [@ambrosiogiglisavare Chapter 8]. Accordingly, we may choose $v^\varepsilon$ to be minimal with respect to the $L^2$ norm, and it will correspond to a gradient vector field $\nabla\phi_\varepsilon$ for some functions $\phi_\varepsilon\colon {\mathbb{R}}^n\to{\mathbb{R}}$, $0<\varepsilon<1$.

The (non-unique) family $(v^\varepsilon)_{0<\varepsilon<1}\in \mathcal V(\mu)$ is the object that we have been pursuing, as it gives precise meaning to the notion of “direction of 