the quantities $\rho$, $w$, and $u$ cannot assume any arbitrary values. In this paper, we assume that there is a unique stagnation density $\rho_\text{max}$, at which vehicles come to a stop, independent of their empty road velocity, i.e., $V(\rho_\text{max},w) = 0$ for all $w$. We therefore have $\rho\in [0,\rho_\text{max})$ and $0 < u\le w$, where the latter inequality follows from ${\frac{\partialV}{\partial\rho}}(\rho,w) < 0$. Moreover, we assume that there is a minimum and maximum empty road velocity, i.e., $0<w_\text{min}\le w\le w_\text{max}$. Because ${\frac{\partialV}{\partial\rho}} < 0$ and ${\frac{\partialV}{\partialw}} > 0$, the function $V(\rho,w)$ can be “inverted” to define the functions (and their domains): $$\begin{aligned}
V:\mathcal{D}_V \longrightarrow [0,w_\text{max}]
&\;\,\text{where~~}
\mathcal{D}_V = \{(\rho,w)\;|\; 0\le\rho<\rho_\text{max},\;
w_\text{min}\le w\le w_\text{max}\}\;, \\
R:\mathcal{D}_R \longrightarrow [0,\rho_\text{max})
&\;\,\text{where~~}
\mathcal{D}_R = \{(u,w)\;|\; 0<u\le w,\; w_\text{min}\le w\le w_\text{max}\}\;, \\
W:\mathcal{D}_W\! \longrightarrow [w_\text{min},w_\text{max}]
&\;\,\text{where~~}
\mathcal{D}_W\! = \{(\rho,u)\;|\; 0\le\rho<\rho_\text{max},\,
V(\rho,w_\text{min})\le u\le V(\rho,w_\text{max})\}\;.\end{aligned}$$ Here the functions $R$ and $W$ are defined as the unique solutions to the problems:

1.  given $u$ and $w$, find $\rho = R(u,w)$, s.t. $V(\rho,w) = u$;

2.  given $\rho$ and $u$, find $w = W(\rho,u)$, s.t. $V(\rho,w) = u$.

From the fact that the quantity $w$ is transported with the flow (while possibly relaxing to some $w_\text{eq} \in (w_\text{min},w_\text{max})$), and 