Relaxation Schemes for Interacting Exclusions

Christophe Bahadoran (Universite Blaise Pascal Clermont-Ferrand)
Jozsef Fritz (Budapest University of Technology and Economics)
Katalin Nagy (Budapest University of Technology and Economics)


We investigate the interaction of one-dimensional asymmetric exclusion processes of opposite speeds, where the exchange dynamics is combined with a creation-annihilation mechanism, and this asymmetric law is regularized by a nearest neighbor stirring of large intensity. The model admits hyperbolic (Euler) scaling, and we are interested in the hydrodynamic behavior of the system in a regime of shocks on the innite line. This work is a continuation of a previous paper by Fritz and Nagy [FN06], where this question has been left open because of the lack of a suitable logarithmic Sobolev inequality. The problem is solved by extending the method of relaxation schemes to this stochastic model, the resulting a priory bound allows us to verify compensated compactness.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 230-262

Publication Date: January 25, 2011

DOI: 10.1214/EJP.v16-857


  1. Bahadoran, Christophe. Blockage hydrodynamics of one-dimensional driven conservative systems. Ann. Probab. 32 (2004), no. 1B, 805--854. MR2039944 (2005f:60209)
  2. Chen, Gui Qiang; Levermore, C. David; Liu, Tai-Ping. Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994), no. 6, 787--830. MR1280989 (95h:35133)
  3. Chen, Gui Qiang; Liu, Tai-Ping. Zero relaxation and dissipation limits for hyperbolic conservation laws. Comm. Pure Appl. Math. 46 (1993), no. 5, 755--781. MR1213992 (94b:35167)
  4.  Chen, Gui-Qiang; Rascle, Michel. Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal. 153 (2000), no. 3, 205--220. MR1771520 (2001f:35250)
  5. Dafermos, Constantine M. Hyperbolic conservation laws in continuum physics. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2005. xx+626 pp. ISBN: 978-3-540-25452-2; 3-540-25452-8 MR2169977 (2006d:35159)
  6. Fritz, J. On the diffusive nature of entropy flow in infinite systems: remarks to a paper: "Nonlinear diffusion limit for a system with nearest neighbor interactions'' [Comm. Math. Phys. 118 (1988), no. 1, 31--59; MR0954674 (89m:60255)] by M. Z. Guo, G. C. Papanicolau and S. R. S. Varadhan. Comm. Math. Phys. 133 (1990), no. 2, 331--352. MR1090429 (92h:60151)
  7. Fritz, J. An Introduction to the Theory of Hydrodynamic Limits. Lectures in Mathematical Sciences 18. The University of Tokyo, ISSN 0919-8180, Tokyo 2001. Math. Review number not available.
  8. Fritz, József. Entropy pairs and compensated compactness for weakly asymmetric systems. Stochastic analysis on large scale interacting systems, 143--171, Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, 2004. MR2073333 (2005f:60213)
  9.  Fritz, József. Microscopic theory of isothermal elasticity. Math. To appear in Arch. Ration. Mech. Anal. Review number not available.
  10. Fritz, József; Nagy, Katalin. On uniqueness of the Euler limit of one-component lattice gas models. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 367--392. MR2285732 (2008k:60244)
  11. Fritz, József; Tóth, Bálint. Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas. Comm. Math. Phys. 249 (2004), no. 1, 1--27. MR2077251 (2005f:82063)
  12. Gosse, Laurent; Tzavaras, Athanasios E. Convergence of relaxation schemes to the equations of elastodynamics. Math. Comp. 70 (2001), no. 234, 555--577 (electronic). MR1813140 (2002f:65128)
  13. Guo, M. Z.; Papanicolaou, G. C.; Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 (1988), no. 1, 31--59. MR0954674 (89m:60255)
  14. Kipnis, Claude; Landim, Claudio. Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. Springer-Verlag, Berlin, 1999. xvi+442 pp. ISBN: 3-540-64913-1 MR1707314 (2000i:60001)
  15. Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4 MR0776231 (86e:60089)
  16. Liu, Tai-Ping. Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987), no. 1, 153--175. MR0872145 (88f:35092) 
  17. Hsiao, Ling; Liu, Tai-Ping. Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Comm. Math. Phys. 143 (1992), no. 3, 599--605. MR1145602 (93e:35072)
  18. Olla, Stefano; Varadhan, S. R. S. Scaling limit for interacting Ornstein-Uhlenbeck processes. Comm. Math. Phys. 135 (1991), no. 2, 355--378. MR1087388 (92h:60154)
  19. Rezakhanlou, Fraydoun. Hydrodynamic limit for attractive particle systems on ${bf Z}sp d$ . Comm. Math. Phys. 140 (1991), no. 3, 417--448. MR1130693 (93f:82035)
  20. Serre, Denis. Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems. Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000. xii+269 pp. ISBN: 0-521-63330-3 MR1775057 (2001c:35146)
  21. Tartar, L. Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136--212, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979. MR0584398 (81m:35014)
  22. Tóth, Bálint; Valkó, Benedek. Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. J. Statist. Phys. 112 (2003), no. 3-4, 497--521. MR1997260 (2004g:82094)
  23. Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions. II. Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), 75--128, Pitman Res. Notes Math. Ser., 283, Longman Sci. Tech., Harlow, 1993. MR1354152 (97a:60144)
  24.  Yau, Horng-Tzer. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 (1991), no. 1, 63--80. MR1121850 (93e:82035)
  25. Yau, Horng-Tzer. Logarithmic Sobolev inequality for generalized simple exclusion processes. Probab. Theory Related Fields 109 (1997), no. 4, 507--538. MR1483598 (99f:60171)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.