2013
DOI: 10.1137/120869729
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Hyperbolic versus Parabolic Asymptotics in Kinetic Theory toward Fluid Dynamic Models

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Cited by 13 publications
(15 citation statements)
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“…Before ending this section, let us briefly introduce the above-mentioned macroscopic systems [17,41,42] of flocking dynamics that have been derived in the literature by means of rigorous hydrodynamic limits. In addition, we will sketch the main results concerning the flocking behavior at large times in some of those models and will compare the kind of scaling proposed by the preceding authors with those considered in this paper and those in [11,50,58] for the Vlasov-Poisson-Fokker-Planck system. Recall that the choice of a Dirac delta in the Mostch-Tadmor alignment operator (14) leads to what authors call strong local alignment, that might also be understood as a relaxation term towards the mean velocity field, i.e., a linear friction term centered at u. Specifically,…”
Section: Macroscopic Systemsmentioning
confidence: 71%
See 1 more Smart Citation
“…Before ending this section, let us briefly introduce the above-mentioned macroscopic systems [17,41,42] of flocking dynamics that have been derived in the literature by means of rigorous hydrodynamic limits. In addition, we will sketch the main results concerning the flocking behavior at large times in some of those models and will compare the kind of scaling proposed by the preceding authors with those considered in this paper and those in [11,50,58] for the Vlasov-Poisson-Fokker-Planck system. Recall that the choice of a Dirac delta in the Mostch-Tadmor alignment operator (14) leads to what authors call strong local alignment, that might also be understood as a relaxation term towards the mean velocity field, i.e., a linear friction term centered at u. Specifically,…”
Section: Macroscopic Systemsmentioning
confidence: 71%
“…Consequently, we recover an analogous convergence result to Corollary 3.13 for any exponent λ ∈ (0, 1/2] and the same limiting macroscopic system (31), hence (1). (42) with γ ∈ (0, 1] differs from that in the Vlassov-Poisson-Fokker-Planck system [11,58]. Specifically, in this paper the velocity diffusion is of low order of ε and each term in the Fokker-Planck differential operator is scaled in a different way (recall the choice V = ε γ in Appendix A).…”
Section: Other Relevant Hydrodynamic Limitsmentioning
confidence: 99%
“…For the readers' convenience, let us mention another alternative to the above hydrodynamic limits in wich the scaling lead to a vanishing inertia effect on the macroscopic limit, thus reducing the second order dynamics to the Smoluchoski first order dynamics. In particular, this line has been developed in 29,110,175,185 for the Vlasov-Poisson-Fokker-Planck system, that give rise to the aggregation equation with Newtonian interactions, i.e.,…”
Section: Hydrodynamic Limits For Lipschitz Influence Functionmentioning
confidence: 99%
“…Therefore, the sequence f n is 1-periodic with respect to x and supported on [0, 1] 2 with respect to (v, u). In order prove that its limit is a solution of (15), we start from the weak formulation of (20); let a test function…”
Section: The Iterative Procedures For the Full Traffic Modelmentioning
confidence: 99%
“…We use now standard results for transport equations, for example, the results in Reference [20] to deduce the convergence of the product measure,…”
Section: The Iterative Procedures For the Full Traffic Modelmentioning
confidence: 99%