Abstract
We investigate the derivation of semilinear relaxation systems and scalar conservation laws from a class of stochastic interacting particle systems. These systems are Markov jump processes set on a lattice, they satisfy detailed mass balance (but not detailed balance of momentum), and are equipped with multiple scalings. Using a combination of correlation function methods with compactness and convergence properties of semidiscrete relaxation schemes we prove that, at a mesoscopic scale, the interacting particle system gives rise to a semilinear hyperbolic system of relaxation type, while at a macroscopic scale it yields a scalar conservation law. Rates of convergence are obtained in both scalings.
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REFERENCES
A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, Lecture Notes in Mathematics 1501 (Springer-Verlag, New York, 1991).
M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law,Comm. Partial Differential Equations 22:195-233 (1997).
K. Uchyiama, On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation, J. Stat. Phys. 52:331-355 (1988).
S. Caprino, A. De Masi, E. Presutti, and M. Pulvirenti, A stochastic particle system modeling the Carleman equation, J. Stat. Phys. 55:625-638 (1989).
S. Caprino, A. De Masi, E. Presutti, and M. Pulvirenti, A derivation of the Broadwell equation, Comm. Math. Phys. 135:443-465 (1991).
S. Caprino and M. Pulvirenti, The Boltzmann_Grad limit for a one-dimensional Boltzmann equation in a stationary state, Comm. Math. Phys. 177:63-81 (1996).
F. Rezakhanlou and J. Tarver, Boltzmann-Grad limit for a particle system in continuum, Ann. Inst. H. Poincaré Probab. Stat. 33:753-796 (1997).
T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108:153-175 (1987).
G.-Q. Chen, C. D. Levermore, and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47:789-830 (1994).
R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49:795-823 (1996).
R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws (1996), preprint.
S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48:235-277 (1995).
A. Tveito and R. Winther, On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term, SIAM J. Math. Anal. 28:136-161 (1997).
M. A. Katsoulakis, G. Kossioris, and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Comm. Partial Differential Equations 24:395-424 (1999).
F. Rezakhanlou, Hydrodynamic limit for attractive particle systems in ℤd, Comm. Math. Phys. 140:417-448 (1991).
B. Perthame and M. Pulvirenti, On some large systems of random particles which approximate scalar conservation laws, Asymptotic Anal. 10:253-278 (1995).
L. Bonaventura, Interface dynamics in an interacting spin system, Nonlinear Anal. 25:799-819 (1995).
M. A. Katsoulakis and P. E. Souganidis, Interacting particle systems and generalized evolution of fronts, Arch. Rational Mech. Anal. 127:133-157 (1994).
P. L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of scalar mulltidimensional conservation laws, J. AMS 7:169-191 (1994).
O. E. Lanford, Time evolution of large classical systems, Lect. Notes in Phys. 38:1-111 (1975).
S. N. Kruzhkov, First order quasilinear equations with several independent variables, Math. USSR Sbornik 10:217-243 (1970).
N. N. Kuznetzov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation, USSR Comp. Math. and Math. Phys. 16:105-119 (1976).
F. Bouchut and B. Perthame, Kruzhkov's estimates for scalar conservation laws revisited, Transactions AMS (1998), to appear.
M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and interacting particles for scalar conservation laws, C. R. Acad. Sci. Paris, Sér. I Math. 323:865-870 (1996).
J. Doob, Stochastic Processes (Wiley, New York, 1953).
T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).
W. Ruijgrok and T. T. Wu, A completely solvable model of the nonlinear Boltzmann equation, Physica A 113:401-416 (1982).
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, New York, 1991).
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Katsoulakis, M.A., Tzavaras, A.E. Multiscale Analysis for Interacting Particles: Relaxation Systems and Scalar Conservation Laws. Journal of Statistical Physics 96, 715–763 (1999). https://doi.org/10.1023/A:1004670308361
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DOI: https://doi.org/10.1023/A:1004670308361