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### Abstract

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics.

We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of traveling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form $({v}_{s},{u}_{s})(t,x-\tilde{x}\left(t\right))$, where $x<\tilde{x}\left(t\right)$ is the congested zone and $x>\tilde{x}\left(t\right)$ is the non-congested zone. The set $\{x=\tilde{x}\left(t\right)\}$ is a free boundary, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a traveling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that traveling waves are asymptotically stable.

### References

[BBCR00] A hierarchy of models for two-phase flows, J. Nonlinear Sci., Volume 10 (2000) no. 6, pp. 639-660 | DOI | Zbl

[BD06] On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl., Volume 86 (2006) no. 4, pp. 362-368 | DOI | Zbl

[BG17] Particle approximation of a constrained model for traffic flow, NoDEA, Nonlinear Differ. Equ. Appl., Volume 24 (2017) no. 5, 55 | Zbl

[BL22] Freely floating objects on a fluid governed by the Boussinesq equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 39 (2022) no. 3, pp. 575-646 | DOI | Zbl

[BLM21] Waves interacting with a partially immersed obstacle in the Boussinesq regime, Anal. PDE, Volume 14 (2021) no. 4, pp. 1085-1124 | DOI | Zbl

[Boc20] Floating structures in shallow water: local well-posedness in the axisymmetric case, SIAM J. Math. Anal., Volume 52 (2020) no. 1, pp. 306-339 | DOI | Zbl

[BPZ14] Singular limit of a Navier–Stokes system leading to a free/congested zones two-phase model, C. R. Math., Volume 352 (2014) no. 9, pp. 685-690 | DOI | Numdam | Zbl

[BR17] Development of congestion in compressible flow with singular pressure, Asymptotic Anal., Volume 103 (2017) no. 1-2, pp. 95-101 | DOI | Zbl

[CGS16] The Compressible to Incompressible Limit of One Dimensional Euler Equations: The Non Smooth Case, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 2, pp. 701-718 | DOI | Zbl

[DHN11] Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., Volume 230 (2011) no. 22, pp. 8057-8088 | DOI | Zbl

[DP20] Existence and stability of partially congested propagation fronts in a one-dimensional Navier–Stokes model, Commun. Math. Sci., Volume 18 (2020) no. 7, pp. 1775-1813 | DOI | Zbl

[Goo86] Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., Volume 95 (1986), pp. 325-344 | DOI | Zbl

[GPSMW18] Congested shallow water model: roof modeling in free surface flow, ESAIM, Math. Model. Numer. Anal., Volume 52 (2018) no. 5, pp. 1679-1707 | DOI | Numdam | Zbl

[Has18] Vortex solutions for the compressible Navier–Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density (2018) (preprint HAL hal-01716150, https://hal.science/hal-01716150/)

[IL19] Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., Volume 70 (2019), pp. 353-464 | DOI | Zbl

[Lan17] On the dynamics of floating structures, Ann. PDE, Volume 3 (2017) no. 1, 11 | Zbl

[LM99] On a free boundary barotropic model, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 16 (1999) no. 3, pp. 373-410 | DOI | Numdam | Zbl

[LSU68] Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968 (Translated from the Russian by S. Smith.) | DOI | Zbl

[Mau12] Prise en compte de la congestion dans les modèles de mouvements de foules, Actes des colloques “EDP-Normandie”, Caen 2010 – Rouen 2011 (Dogbe, Christian et al., eds.) (Normandie-Mathématiques), Fédération Normandie-Mathématiques (2012), pp. 7-20 | Zbl

[MRCSV11] Handling congestion in crowd motion modeling, Netw. Heterog. Media, Volume 6 (2011) no. 3, pp. 485-519 | DOI | Zbl

[MSMTT19] Analysis of a simplified model of rigid structure floating in a viscous fluid, J. Nonlinear Sci., Volume 29 (2019) no. 5, pp. 1975-2020 | DOI | Zbl

[PS22] Numerical Staggered Schemes for the Free-Congested Navier–Stokes Equations, SIAM J. Numer. Anal., Volume 60 (2022) no. 4, pp. 1824-1852 | DOI | Zbl

[PZ15] Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier–Stokes equations, Commun. Partial Differ. Equations, Volume 40 (2015) no. 8, pp. 1558-1589 | DOI | Zbl

[She84] On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas, J. Appl. Math. Stochastic Anal., Volume 48 (1984) no. 6, pp. 665-672 | Zbl

[Shi16] On the $\mathcal{R}$-Boundedness for the Two Phase Problem with Phase Transition: Compressible-Incompressible Model Problem, Funkc. Ekvacioj, Ser. Int., Volume 59 (2016) no. 2, pp. 243-287 | DOI | Zbl

[VY16] Nonlinear stability of viscous shock wave to one-dimensional compressible isentropic Navier–Stokes equations with density dependent viscous coefficient, Commun. Math. Sci., Volume 14 (2016) no. 8, pp. 2215-2228 | DOI | Zbl

[Wag87] Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differ. Equations, Volume 68 (1987) no. 1, pp. 118-136 | DOI | Zbl