51 research outputs found
Strong Convergence towards self-similarity for one-dimensional dissipative Maxwell models
We prove the propagation of regularity, uniformly in time, for the scaled
solutions of one-dimensional dissipative Maxwell models. This result together
with the weak convergence towards the stationary state proven by Pareschi and
Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1
norm towards it depending on the regularity of the initial data. In the case of
the one-dimensional inelastic Boltzmann equation, the result does not depend of
the degree of inelasticity. This generalizes a recent result of Carlen,
Carrillo and Carvalho (arXiv:0805.1051v1), in which, for weak inelasticity,
propagation of regularity for the scaled inelastic Boltzmann equation was found
by means of a precise control of the growth of the Fisher information.Comment: 26 page
Structure preserving schemes for mean-field equations of collective behavior
In this paper we consider the development of numerical schemes for mean-field
equations describing the collective behavior of a large group of interacting
agents. The schemes are based on a generalization of the classical Chang-Cooper
approach and are capable to preserve the main structural properties of the
systems, namely nonnegativity of the solution, physical conservation laws,
entropy dissipation and stationary solutions. In particular, the methods here
derived are second order accurate in transient regimes whereas they can reach
arbitrary accuracy asymptotically for large times. Several examples are
reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic
Problem
Collisional rates for the inelastic Maxwell model: application to the divergence of anisotropic high-order velocity moments in the homogeneous cooling state
The collisional rates associated with the isotropic velocity moments
and
are exactly derived in the case of the
inelastic Maxwell model as functions of the exponent , the coefficient of
restitution , and the dimensionality . The results are applied to
the evolution of the moments in the homogeneous free cooling state. It is found
that, at a given value of , not only the isotropic moments of a degree
higher than a certain value diverge but also the anisotropic moments do. This
implies that, while the scaled distribution function has been proven in the
literature to converge to the isotropic self-similar solution in well-defined
mathematical terms, nonzero initial anisotropic moments do not decay with time.
On the other hand, our results show that the ratio between an anisotropic
moment and the isotropic moment of the same degree tends to zero.Comment: 7 pages, 2 figures; v2: clarification of some mathematical statements
and addition of 7 new references; v3: Published in "Special Issue: Isaac
Goldhirsch - A Pioneer of Granular Matter Theory
Uncertainty quantification for kinetic models in socio-economic and life sciences
Kinetic equations play a major rule in modeling large systems of interacting
particles. Recently the legacy of classical kinetic theory found novel
applications in socio-economic and life sciences, where processes characterized
by large groups of agents exhibit spontaneous emergence of social structures.
Well-known examples are the formation of clusters in opinion dynamics, the
appearance of inequalities in wealth distributions, flocking and milling
behaviors in swarming models, synchronization phenomena in biological systems
and lane formation in pedestrian traffic. The construction of kinetic models
describing the above processes, however, has to face the difficulty of the lack
of fundamental principles since physical forces are replaced by empirical
social forces. These empirical forces are typically constructed with the aim to
reproduce qualitatively the observed system behaviors, like the emergence of
social structures, and are at best known in terms of statistical information of
the modeling parameters. For this reason the presence of random inputs
characterizing the parameters uncertainty should be considered as an essential
feature in the modeling process. In this survey we introduce several examples
of such kinetic models, that are mathematically described by nonlinear Vlasov
and Fokker--Planck equations, and present different numerical approaches for
uncertainty quantification which preserve the main features of the kinetic
solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic
Equations
Remarks on the H theorem for a non involutive Boltzmann like kinetic model
In this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation v* = pv + qw, w* = qv + pw, where (v, w) are the pre-collisional velocities and (v*, w*) the post-collisional ones and p 65 q > 0 are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys., 124(2\u20134):747\u2013779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case p2 + q2 = 1, even if the transformation has Jacobian J 60 1 and so it is not involutive, we remark that the H Theorem holds true. As a consequence we prove exponential convergence in L1 of the solution to the stationary state, which is the Maxwellian
Convergence to self-similarity for the Boltzmann equation for strongly inelastic Maxwell molecules
Abstract. We prove propagation of regularity, uniformly in time, for the scaled solution
Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules
We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform
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