An overview on congestion phenomena in fluid equations

Perrin, Charlotte

Journées équations aux dérivées partielles (2018), Exposé no. 6, 34 p.

We review some recent analysis results and open perspectives around congestion phenomena in fluid equations. The PDE systems under study are based on Navier–Stokes equations in which congestion is encoded in a maximal density constraint. The paper is organized around three main topics: multi-scale issues, regularity issues and finally non-locality issues.

Mots clés : Navier–Stokes equations, Euler equations, maximal packing constraint, phase transitions

[1] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Gradient flows: in metric spaces and in the space of probability measures, Lectures in Mathematics, Birkhäuser, 2008 | Zbl

[2] Andreotti, Bruno; Forterre, Yoël; Pouliquen, Olivier Granular media: between fluid and solid, Cambridge University Press, 2013 | Zbl

[3] Ballard, Patrick Formulation and well-posedness of the dynamics of rigid-body systems with perfect unilateral constraints, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., Volume 359 (2001) no. 1789, pp. 2327-2346 | Zbl

[4] Benzoni-Gavage, Sylvie; Serre, Denis Multidimensional hyperbolic partial differential equations, Oxford Mathematical Monographs, Oxford University Press, 2007 | Zbl

[5] Berthelin, Florent Existence and weak stability for a pressureless model with unilateral constraint, Math. Models Methods Appl. Sci., Volume 12 (2002) no. 2, pp. 249-272 | Zbl

[6] Berthelin, Florent Theoretical study of a multidimensional pressureless model with unilateral constraint, SIAM J. Math. Anal., Volume 49 (2017) no. 3, pp. 2287-2320

[7] Berthelin, Florent; Broizat, Damien A model for the evolution of traffic jams in multi-lane, Kinet. Relat. Models, Volume 5 (2012) no. 4, pp. 697-728 | Zbl

[8] Berthelin, Florent; Degond, Pierre; Delitala, Marcello; Rascle, Michel A model for the formation and evolution of traffic jams, Arch. Ration. Mech. Anal., Volume 187 (2008) no. 2, pp. 185-220 | Zbl

[9] Bocchi, Edoardo Floating structures in shallow water: local well-posedness in the axisymmetric case (2018) (https://arxiv.org/abs/1802.07643)

[10] Bouchut, François On zero pressure gas dynamics, Advances in kinetic theory and computing: selected papers (Series on Advances in Mathematics for Applied Sciences), Volume 22, World Scientific, 1994, pp. 171-190 | Zbl

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

[12] Bourdarias, Christian; Ersoy, Mehmet; Gerbi, Stéphane A mathematical model for unsteady mixed flows in closed water pipes, Sci. China, Math., Volume 55 (2012) no. 2, pp. 221-244 | Zbl

[13] Bouzid, Mehdi; Izzet, Adrien; Trulsson, Martin; Clément, Eric; Claudin, Philippe; Andreotti, Bruno Non-local rheology in dense granular flows, Eur. Phys. J. E, Volume 38 (2015) no. 11, 125 pages | DOI

[14] Brenier, Yann; Gangbo, Wilfrid; Savaré, Giuseppe; Westdickenberg, Michael Sticky particle dynamics with interactions, J. Math. Pures Appl., Volume 99 (2013) no. 5, pp. 577-617 | Zbl

[15] Brenier, Yann; Grenier, Emmanuel Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., Volume 35 (1998) no. 6, pp. 2317-2328 | Zbl

[16] 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 | Zbl

[17] Bresch, Didier; Necasova, Sarka; Perrin, Charlotte Compression effects in heterogeneous media (2018) (https://arxiv.org/abs/1807.06360)

[18] 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. Acad. Sci. Paris, Volume 352 (2014) no. 9, pp. 685-690 | Zbl

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

[20] Cavalletti, Fabio; Sedjro, Marc; Westdickenberg, Michael A simple proof of global existence for the 1d pressureless gas dynamics equations, SIAM J. Math. Anal., Volume 47 (2015) no. 1, pp. 66-79 | Zbl

[21] Cazacu, Oana; Ionescu, Ioan R.; Perrot, Thomas Steady-state flow of compressible rigid–viscoplastic media, Int. J. Eng. Sci., Volume 44 (2006) no. 15-16, pp. 1082-1097 | Zbl

[22] Dalibard, Anne-Laure; Perrin, Charlotte Existence and stability of partially congested propagation fronts in a one-dimensional Navier–Stokes model (2019) (https://arxiv.org/abs/1902.02982)

[23] 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

[24] Degond, Pierre; Minakowski, Piotr; Navoret, Laurent; Zatorska, Ewelina Finite volume approximations of the Euler system with variable congestion, Comput. Fluids, Volume 169 (2017), pp. 23-39 | Zbl

[25] Degond, Pierre; Minakowski, Piotr; Zatorska, Ewelina Transport of congestion in two-phase compressible/incompressible flows, Nonlinear Anal., Real World Appl., Volume 42 (2018), pp. 485-510 | Zbl

[26] Denisova, Irina; Solonnikov, Vsevolod Local and global solvability of free boundary problems for the compressible Navier–Stokes equations near equilibria, Handbook of mathematical analysis in mechanics of viscous fluids (Springer Reference), Springer, 2018, pp. 1-88 | Zbl

[27] Feireisl, Eduard Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, 2004 | Zbl

[28] Feireisl, Eduard; Jin, Bum Ja; Novotný, Antonín Relative entropies, suitable weak solutions and weak-strong uniqueness for the compressible Navier–Stokes system, J. Math. Fluid Mech., Volume 14 (2012) no. 4, pp. 717-730 | Zbl

[29] Feireisl, Eduard; Lu, Yong; Málek, Josef On PDE analysis of flows of quasi-incompressible fluids, ZAMM, Z. Angew. Math. Mech., Volume 96 (2016) no. 4, pp. 491-508 | DOI

[30] Feireisl, Eduard; Lu, Yong; Novotný, Antonín Weak-strong uniqueness for the compressible Navier–Stokes equations with a hard-sphere pressure law, Sci. China Math., Volume 61 (2018) no. 11, pp. 2003-2016 | DOI

[31] Feireisl, Eduard; Novotný, Antonín Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Springer, 2009 | Zbl

[32] Fuchs, Martin; Seregin, Gregory Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lecture Notes in Mathematics, 1749, Springer, 2000 | Zbl

[33] Godlewski, Edwige; Parisot, Martin; Sainte-Marie, Jacques; Wahl, Fabien Congested shallow water model: floating object (2018) (https://hal.inria.fr/hal-01871708)

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

[35] Hecht, Sophie; Vauchelet, Nicolas Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint, Commun. Math. Sci., Volume 15 (2017) no. 7, pp. 1913-1932 | Zbl

[36] Iguchi, Tatsuo; Lannes, David Hyperbolic free boundary problems and applications to wave-structure interactions (2018) (https://arxiv.org/abs/1806.07704)

[37] Labbé, Stéphane; Maitre, Emmanuel A free boundary model for Korteweg fluids as a limit of barotropic compressible Navier–Stokes equations, Methods and Applications of Analysis, Volume 20 (2013) no. 2, pp. 165-178 | DOI

[38] Lannes, David On the dynamics of floating structures, Ann. PDE, Volume 3 (2017) no. 1, 11, 81 pages | Zbl

[39] Lefebvre, Aline Modélisation numérique d’écoulements fluide-particules: prise en compte des forces de lubrification, Université Paris Sud - Paris XI (France) (2007) (Ph. D. Thesis)

[40] Lefebvre, Aline Numerical simulation of gluey particles, ESAIM, Math. Model. Numer. Anal., Volume 43 (2009) no. 1, pp. 53-80 | Zbl

[41] Lefebvre-Lepot, Aline; Maury, Bertrand Micro-macro modelling of an array of spheres interacting through lubrication forces, Adv. Math. Sci. Appl., Volume 21 (2011) no. 2, pp. 535-557 | Zbl

[42] Lions, Pierre-Louis Mathematical topics in fluid mechanics. Vol. 1: Incompressible models, Oxford Lecture Series in Mathematics and its Applications, 3, Oxford University Press, 1996 | Zbl

[43] Lions, Pierre-Louis Mathematical topics in fluid mechanics: Vol. 2: Compressible Models, Oxford Lecture Series in Mathematics and its Applications, 10, Oxford University Press, 1998 | Zbl

[44] 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 | Zbl

[45] Málek, Josef; Nečas, Jindřich; Rajagopal, Kumbakonam Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal., Volume 165 (2002) no. 3, pp. 243-269 | Zbl

[46] Málek, Josef; Rajagopal, Kumbakonam Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities, Handbook of mathematical fluid dynamics 4, Elsevier, 2007, pp. 407-444 | DOI

[47] Marly, Arthur Analyse mathématique et numérique d’écoulements de fluides à seuil, École Normale Supérieure de Lyon (France) (2018) (Ph. D. Thesis)

[48] Maury, Bertrand A gluey particle model, ESAIM, Proc., Volume 18 (2007), pp. 133-142 | Zbl

[49] Maury, Bertrand Prise en compte de la congestion dans les modeles de mouvements de foules, Actes des colloques EDP-Normandie (Caen 2010 - Rouen 2011, Fédération Normandie-Mathématiques, 2012, pp. 7-20 | Zbl

[50] Maury, Bertrand; Roudneff-Chupin, Aude; Santambrogio, Filippo A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 10, pp. 1787-1821 | Zbl

[51] 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 | Zbl

[52] Mellet, A.; Vasseur, A. On the barotropic compressible Navier–Stokes equations, Commun. Partial Differ. Equations, Volume 32 (2007) no. 3, pp. 431-452 | DOI

[53] Natile, Luca; Savaré, Giuseppe A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal., Volume 41 (2009) no. 4, pp. 1340-1365 | Zbl

[54] Nelson, Edward Dynamical theories of Brownian motion, Mathematical Notes, Princeton University Press, 1967 | Zbl

[55] Novotný, Antonín; Straškraba, Ivan Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications, 27, Oxford University Press, 2004 | Zbl

[56] Perepelitsa, Mikhail On the global existence of weak solutions for the Navier–-Stokes equations of compressible fluid flows, SIAM J. Math. Anal., Volume 38 (2006) no. 4, pp. 1126-1153 | Zbl

[57] Perrin, Charlotte Pressure-dependent viscosity model for granular media obtained from compressible Navier–Stokes equations, AMRX, Appl. Math. Res. Express, Volume 2016 (2016) no. 2, pp. 289-333 | Zbl

[58] Perrin, Charlotte Modelling of phase transitions in one-dimensional granular flows, ESAIM, Proc. Surv., Volume 58 (2017), pp. 78-97 | Zbl

[59] Perrin, Charlotte; Westdickenberg, M. One-dimensional granular system with memory effects, SIAM J. Math. Anal., Volume 50 (2018) no. 6, pp. 5921-5946 | Zbl

[60] 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 | Zbl

[61] Perthame, Benoît; Quirós, Fernando; Vázquez, Juan Luis The Hele–Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal., Volume 212 (2014) no. 1, pp. 93-127 | Zbl

[62] Perthame, Benoît; Vauchelet, Nicolas Incompressible limit of a mechanical model of tumor growth with viscosity, Philos. Trans. A, R. Soc. Lond., Volume 373 (2015) no. 2050, 2014283, 16 pages | Zbl

[63] Pouliquen, Olivier; Forterre, Yoël A non-local rheology for dense granular flows, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., Volume 367 (2009) no. 1909, pp. 5091-5107 | Zbl

[64] Preux, Anthony Transport optimal et équations des gaz sans pression avec contrainte de densité maximale, Université Paris-Saclay (France) (2016) (Ph. D. Thesis)

[65] Renardy, Michael Some remarks on the Navier–Stokes equations with a pressure-dependent viscosity, Commun. Partial Differ. Equations, Volume 11 (1986), pp. 779-793 | Zbl

[66] Serre, Denis Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves, Cambridge University Press, 1999 | Zbl

[67] 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 | Zbl

[68] Vaigant, V. A.; Kazhikhov, Alexandre V. On existence of global solutions to the two-dimensional Navier–Stokes equations for a compressible viscous fluid, Sib. Math. J., Volume 36 (1995) no. 6, pp. 1108-1141 | Zbl

[69] Vauchelet, Nicolas; Zatorska, Ewelina Incompressible limit of the Navier–Stokes model with a growth term, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 163 (2017), pp. 34-59 | Zbl