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Propagation trajectories of chaos for a scalar conservation law. (Propagation trajectorielle du chaos pour les lois de conservation scalaire.) (French) Zbl 0913.60094

Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 215-230 (1998).
The author studies a kinetic nonlinear equation introduced by B. Perthame and E. Tadmor [Commun. Math. Phys. 136, No. 3, 501-517 (1991; Zbl 0729.76070)], in a probabilistic point of view. If the initial condition is integrable, the equation admits a unique weak solution in \(L^\infty([0,T], L^1)\). In the first part, the author gives a probabilistic interpretation of this solution when the initial data is a density of probability. He introduces a nonlinear martingale problem associated with the equation as follows. The marginals at time \(t\) of each solution admit densities with respect to the Lebesgue measure, which are weak solutions of the nonlinear equation. He shows the existence and uniqueness of the solution of the martingale problem. In the second part, the author generalizes a propagation of chaos result obtained by B. Perthame and M. Pulvirenti [Asymptotic Anal. 10, No. 3, 263-278 (1995; Zbl 0846.35081)] for fixed times to a pathwise result in variation norm. By the probabilistic approach, the hypotheses on the initial assumption are weaker. The proof is based on a coupling between interacting particles and independent particles associated with the nonlinear martingale problem.
For the entire collection see [Zbl 0893.00035].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
35K55 Nonlinear parabolic equations
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