Local and global well-posedness of one-dimensional free-congested equations

Dalibard, Anne-Laure; Perrin, Charlotte

Annales Henri Lebesgue, Tome 7 (2024), pp. 1175-1243

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.

Cet article est consacré à l’étude d’un modèle de congestion unidimensionnel, composé de deux phases distinctes. Dans la phase congestionnée, la pression est libre et la dynamique est incompressible, tandis que dans la phase non congestionnée, le fluide obéit à une dynamique compressible avec une pression nulle.

Nous étudions le problème de Cauchy pour des données initiales qui sont de petites perturbations de profils d’ondes progressives dans la zone non congestionnée. Nous prouvons deux résultats différents. Premièrement, nous montrons que pour des perturbations arbitrairement grandes, le problème de Cauchy est localement bien posé dans des espaces de Sobolev à poids. La solution obtenue prend la forme $(v_s,u_s)(t,x-\tilde{x}\left(t\right))$, où $x<\tilde{x}\left(t\right)$ est la zone congestionnée et $x>\tilde{x}\left(t\right)$ est la zone libre. La ligne $\{x=\tilde{x}\left(t\right)\}$ est une frontière libre, dont l’évolution est couplée à celle de la solution. Deuxièmement, nous prouvons que si la perturbation initiale est suffisamment petite, alors la solution est globale. Ce résultat de stabilité repose sur des propriétés de coercivité de l’opérateur linéarisé autour d’une onde progressive, et sur l’introduction d’une nouvelle inconnue qui satisfait de meilleures estimations que l’inconnue initiale. Dans ce cas, nous prouvons également que les ondes progressives sont asymptotiquement stables.

Classification : 35Q35, 35L67
Keywords: Navier–Stokes equations, free boundary problem, nonlinear stability

[BBCR00] Bouchut, François; Brenier, Yann; Cortes, Julien; Ripoll, Jean-François A hierarchy of models for two-phase flows, J. Nonlinear Sci., Volume 10 (2000) no. 6, pp. 639-660 | MR | Zbl | DOI

[BD06] Bresch, Didier; Desjardins, Benoît 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 | MR | Zbl | DOI

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

[BL22] Beck, Geoffrey; Lannes, David 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 | MR | Zbl | DOI

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

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

[BPZ14] Bresch, Didier; Perrin, Charlotte; Zatorska, Ewelina 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 | Zbl | DOI | Numdam | MR

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

[CGS16] Colombo, Rinaldo; Guerra, Graziano; Schleper, Veronika 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 | Zbl | DOI | MR

[DHN11] Degond, Pierre; Hua, Jiale; Navoret, Laurent Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., Volume 230 (2011) no. 22, pp. 8057-8088 | Zbl | DOI | MR

[DP20] Dalibard, Anne-Laure; Perrin, Charlotte 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 | Zbl | DOI | MR

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

[GPSMW18] Godlewski, Edwige; Parisot, Martin; Sainte-Marie, Jacques; Wahl, Fabien Congested shallow water model: roof modeling in free surface flow, ESAIM, Math. Model. Numer. Anal., Volume 52 (2018) no. 5, pp. 1679-1707 | MR | Zbl | DOI | Numdam

[Has18] Haspot, Boris 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] Iguchi, Tatsuo; Lannes, David Hyperbolic free boundary problems and applications to wave-structure interactions, Indiana Univ. Math. J., Volume 70 (2019), pp. 353-464 | MR | Zbl | DOI

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

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

[LSU68] Ladyženskaja, Olga A.; Solonnikov, Vsevolod A.; Ural’tseva, Nina N. Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968 (Translated from the Russian by S. Smith.) | MR | Zbl | DOI

[Mau12] Maury, Bertrand 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] Maury, Bertrand; Roudneff-Chupin, Aude; Santambrogio, Filippo; Venel, Juliette Handling congestion in crowd motion modeling, Netw. Heterog. Media, Volume 6 (2011) no. 3, pp. 485-519 | MR | Zbl | DOI

[MSMTT19] Maity, Debayan; San Martín, Jorge; Takahashi, Takéo; Tucsnak, Marius Analysis of a simplified model of rigid structure floating in a viscous fluid, J. Nonlinear Sci., Volume 29 (2019) no. 5, pp. 1975-2020 | MR | Zbl | DOI

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

[PZ15] Perrin, Charlotte; Zatorska, Ewelina 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 | MR | Zbl | DOI

[She84] Shelukhin, Vladimir V. 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 | MR | Zbl

[Shi16] Shibata, Yoshihiro On the $ℛ$-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 | MR | Zbl | DOI

[VY16] Vasseur, Alexis F.; Yao, Lei 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 | MR | Zbl | DOI

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