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Hydrodynamics of Porous Medium Model with Slow Reservoirs


We analyze the hydrodynamic behavior of the porous medium model (PMM) in a discrete space \(\{0,\ldots , n\}\), where the sites 0 and n stand for reservoirs. Our strategy relies on the entropy method of Guo et al. (Commun Math Phys 118:31–59, 1988). However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations). In order to avoid this problem, we slightly perturbed the dynamics in such a way that the macroscopic behavior of the system keeps following the porous medium equation (PME), but with boundary conditions which depend on the reservoirs’ strength.

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  1. 1.

    Andersen, H., Kob, W.: Kinetic lattice-gas model of cage effects in high-density liquids and a test of mode-coupling theory of the ideal-glass transition. Phys. Rev. E 48, 4359–4363 (1993)

    Google Scholar 

  2. 2.

    Baldasso, R., Menezes, O., Neumann, A., Souza, R.: Exclusion Process with Slow Boundary. J. Stat. Phys. 167(5), 1112–1142 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  3. 3.

    Blondel, O., Gonçalves, P., Simon, M.: Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics. Electron. J. Probab. 2(69), 1–25 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Cancrini, N., Martinelli, F., Roberto, C., Toninelli, C.: Kinetically constrained lattice gases. Commun. Math. Phys. 297(2), 299–344 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  5. 5.

    de Paula, R., Gonçalves, P., Neumann, A.: Porous Medium Model in Contact with Slow Reservoirs. From Particle Systems to Partial Differential Equations, pp. 123–147. Springer, New York (2018)

    MATH  Google Scholar 

  6. 6.

    Derrida, B., Evans, M., Hakim, V., Pasquier, V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26(7), 1493 (1993)

    ADS  MathSciNet  MATH  Google Scholar 

  7. 7.

    Ekhaus, M., Seppalainen, T.: Stochastic dynamics macroscopically governed by the porous medium equation. Annales 21(2), 309–352 (1996)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Evans, L.: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998)

    Google Scholar 

  9. 9.

    Feng, S., Iscoe, I., Seppalainen, T.: A microscopic mechanism for the porous medium equation. Stoch. Process. Appl. 66, 147–182 (1997)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Filo, J.: A nonlinear diffusion equation with nonlinear boundary conditions: methods of lines. Math. Slovaca 38(3), 273–296 (1988)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Gonçalves, P.: Hydrodynamics for symmetric exclusion in contact with reservoirs, Stochastic Dynamics Out of Equilibrium, Institut Henri Poincaré, Paris, France, 2017. Springer Proceedings in Mathematics and Statistics Book Series, vols. 137–205 (2019)

  12. 12.

    Gonçalves, P., Landim, C., Toninelli, C.: Hydrodynamic limit for a particle system with degenerate rates. Ann. Inst. H. Poincaré: Probab. Statist 45(4), 887–909 (2009)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)

    ADS  MathSciNet  MATH  Google Scholar 

  14. 14.

    Gurney, W., Nisbet, R.: The regulation of inhomogeneous populations. J. Theor. Biol. 52, 441–457 (1975)

    Google Scholar 

  15. 15.

    Gurtin, M., MacCarny, R.: On the diffusion of biological populations. Math. Biosci. 33, 35–49 (1977)

    MathSciNet  Google Scholar 

  16. 16.

    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320. Springer, Berlin (1999)

  17. 17.

    Ladyženskaja, A., Solonnikov, A., Ural’ceva, N.: Linear and Quasi-linear Equations of Parabolic Type. Amer. Math. Soc, Providence, RI (1968)

    Google Scholar 

  18. 18.

    Lieberman, M.: Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113, 422–440 (1986)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Muskat, M.: The Flow of Homegeneous Fluids Through Porous Media. McGrawHill, New York (1937)

    MATH  Google Scholar 

  20. 20.

    Ritort, F., Sollich, P.: Glassy dynamics of kinetically constrained models. Adv. Phys. 52(4), 219–342 (2003)

    ADS  Google Scholar 

  21. 21.

    Vazquez, J.: The Porous Medium Equation - Mathematical Theory. Claredon Press, Oxford (2007)

    MATH  Google Scholar 

  22. 22.

    Yau, H.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22(1), 63–80 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  23. 23.

    Zel’dovich, B., Raizer, P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena II. Academic Press, New York (1966)

    Google Scholar 

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A.N. was supported through a Grant “L’ORÉAL - ABC - UNESCO Para Mulheres na Ciência”. R.P. thanks FCT/Portugal for support through the project Lisbon Mathematics PhD (LisMath). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (Grant Agreement No 715734).

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Correspondence to P. Gonçalves.

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Communicated by Stefano Olla.

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Bonorino, L., de Paula, R., Gonçalves, P. et al. Hydrodynamics of Porous Medium Model with Slow Reservoirs. J Stat Phys 179, 748–788 (2020).

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  • Porous medium model
  • Hydrodynamic limit
  • Porous medium equation
  • Boundary conditions