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Journal of Statistical Physics

, Volume 170, Issue 3, pp 466–491 | Cite as

Density Large Deviations for Multidimensional Stochastic Hyperbolic Conservation Laws

  • J. Barré
  • C. BernardinEmail author
  • R. Chetrite
Article

Abstract

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law. When the mobility and diffusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the problem has been solved in Bellettini and Mariani (Bull Greek Math Soc 57:31–45, 2010). When this proportionality does not hold, we compute explicitly the large deviation function for a step-like density profile, and we show that the associated optimal current has a non trivial structure. We also derive a lower bound for the large deviation function, valid for a more general weak solution, and leave the general large deviation function upper bound as a conjecture.

Keywords

Large deviations principle Stochastic conservation laws Kawasaki dynamics Active particles 

Notes

Acknowledgements

We acknowledge very useful discussions with C. Bahadoran, T. Bodineau, M. Mariani and C. Nardini. This work has been supported by the Brazilian-French Network in Mathematics and the project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and the project LSD ANR-15-CE40-0020-01 LSD of the French National Research Agency (ANR). This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program Non-equilibrium statistical physics (Code: ICTS/Prog-NESP/2015/10).

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Copyright information

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Authors and Affiliations

  1. 1.MAPMO - UMR CNRS 7349, Fédération Denis Poisson Université d’Orléans, Collegium Sciences et TechniquesOrléans Cedex 2France
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.Université Côte d’Azur, CNRS, LJADNice Cedex 02France

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