6533b838fe1ef96bd12a3bd6
RESEARCH PRODUCT
Empirical measures and Vlasov hierarchies
Clément MouhotValeria RicciFrançois Golsesubject
MSC 82C05 (35F25 28A33)[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]FOS: Physical sciences01 natural sciencesVlasov type equation Mean-field limit Empirical measure BBGKY hierarchy Monge-Kantorovich distanceMathematics - Analysis of PDEs[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]FOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]Applied mathematicsMonge-Kantorovich distanceDirect proof0101 mathematicsEmpirical measureMathematical PhysicsMean field limitMathematicsNumerical AnalysisHierarchy010102 general mathematicsVlasov type equationMathematical Physics (math-ph)Empirical measureBBGKY hierarchyLipschitz continuity010101 applied mathematicsKernel (algebra)Uniqueness theorem for Poisson's equationBBGKY hierarchyModeling and SimulationExponent82C05 (35F25 28A33)Analysis of PDEs (math.AP)description
The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, Math. Meth. Appl. Sci. 3 (1981), 445-455].
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2013-01-01 | Kinetic & Related Models |