This short note concerns the formal limit presented in [J. Comput Phys230 (2011), 8057–8088] and [Actes colloquia Rouen (2012)] between the compressible Euler equations with singular pressure (soft model) and the pressureless Euler system with unilateral constraint (hard model). These soft and hard models with maximal constraint on the density are used to reproduce congestion phenomena. In this paper, we are interested in the question how the different regions (congested and non-congested regions) fit together and how the transition occurs when congestion first develops in the initial system namely the compressible Euler equation with singular pressure. We shall develop a formal solution by matched asymptotics and show that a shock front may separate the congested region from the outside non-congested region. This, in some sense, shows that to justify the formal limit between the two models (soft and hard Euler systems), the shock fronts formation has to be considered and mathematically analyzed.
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