Soft congestion approximation to the one-dimensional constrained Euler equations

Abstract

This article is concerned with the analysis of the one-dimensional compressible Euler equations with a singular pressure law, the so-called hard sphere equation of state. We provide a detailed description of the effect of the singular pressure on the breakdown of the smooth solutions. Moreover, we rigorously justify the singular limit for smooth solutions towards the free-congested Euler equations, where the compressible (free) dynamics is coupled with the incompressible one in the constrained (i.e. congested) domain.

Hard congestion limit of the dissipative Aw–Rascle system

“…Theorem 2.1. Let ε > 0 be fixed, and let p ε , φ ε be given by (9). Assume that the initial data (11) satisfy…”

“…Consequently, solutions to system (6) are particular solutions to (7). In [4] the constrained Euler equations are obtained through the sticky blocks approximation, while in [1,9,18], the constrained Euler system is approximated by the compressible Euler equations with singular pressure P ε = P ε (ρ ε ). Similar asymptotic limit passage ε → 0 was analysed in the multi-dimensional setting by Bresch, Necasova and Perrin [14] in the case of heterogeneous fluids flows described by compressible Birkmann equations with singular pressure and bulk viscosity coefficient.…”

Hard-congestion limit of the p-system in the BV setting

“…• Let us finally observe that the present setting and the use of the Wave Front Tracking algorithm allows us to "track" the dynamics of the interface for all times t ≥ 0. This is a strong improvement compared to the previous work of Bianchini & Perrin [8] where the limit ε → 0 is tackled by means of compactness methods and a weak (L 1 ) control of the pressure on a limited time interval (independent of ε). The study of global-in-time solutions is also a important difference with the study of Iguchi and Lannes [19] on the floating body problem.…”

“…The isentropic component κϱ γi appearing in (0.2) can be then understood as the hydrostatic pressure in the shallow water equations. From the mathematical viewpoint, the singular limit ε → 0 from soft-congestion systems towards hard congestion systems has been previously studied in various frameworks, in particular: a strong C 1 (local-in-time) setting in [8], and a weak setting (namely global-in-time finite energy weak solutions) when additional viscosity is taken into account, see for instance [24], or at the level of the Riemann problems in [16] (Appendix A). Weak solutions to the Eulerian version of (0.3), when κ = 0, have been constructed by other means: through a discrete (sticky blocks) approximation in [4], via a convex optimization point of view in [23].…”

Hard-congestion limit of the p-system in the BV setting

“…From the mathematical standpoint, the singular limit ε → 0 from soft-congestion systems towards hard congestion systems has been previously studied in various frameworks, in particular: a strong C 1 (local-in-time) setting in [8], and a weak setting (namely global-in-time finite energy weak solutions) when additional viscosity is taken into account, see for instance [23], or at the level of the Riemann problems in [16] (Appendix A). Weak solutions to the Eulerian version of (1.3), when κ = 0, have been constructed by other means: through a discrete (sticky blocks) approximation in [4], via an convex optimization point of view in [22].…”