  We introduce the notion of duality solution for the hard-congestion model on
the real line, and additionally prove an existence result for this class of
solutions. Our study revolves around the analysis of a generalised Aw-Rascle
system, where the offset function is replaced by the gradient of a singular
function, such as $\rho$ $\gamma$ n , where $\gamma$ $\rightarrow$ $\infty$. We
prove that under suitable assumptions on the initial data, solutions to the
Aw-Rascle system converge towards the so-called duality solutions, which have
previously found applications in other systems which exhibit compressive
dynamics. We also prove that one can obtain weak solutions to the limiting
system under stricter assumptions on the initial data. Finally, we discuss
(non-)uniqueness issues.
