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Kinetic & Related Models

2010 , Volume 3 , Issue 1

Special Issue on Nonholonomic Constraints in Mechanics and Optimal Control Theory

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Preface
José Antonio Carrillo, Peter Markowich and Lorenzo Pareschi
2010, 3(1): i-i doi: 10.3934/krm.2010.3.1i +[Abstract](303) +[PDF](27.1KB)
Abstract:
The idea of putting together a special issue of KRM in honor of Giuseppe Toscani's 60$th$ birthday was received with great enthusiasm by the "kinetic" community and by Giuseppe's scientific colleagues and friends from other mathematical communities. There is a common feeling throughout that we have learned a great deal from the ideas, the methodologies and the scientific conclusions of Giuseppe Toscani's work.
   The papers related to Toscani's research in this special issue are authored by collaborators, friends and students of Giuseppe. The topics include classical and non classical applications of kinetic theory, entropy-entropy dissipation methods, asymptotic techniques and numerical simulations.The articles in this volume are ordered alphabetically by names of authors.

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Small velocity and finite temperature variations in kinetic relaxation models
Kazuo Aoki, Ansgar Jüngel and Peter A. Markowich
2010, 3(1): 1-15 doi: 10.3934/krm.2010.3.1 +[Abstract](368) +[PDF](202.2KB)
Abstract:
A small Knuden number analysis of a kinetic equation in the diffusive scaling is performed. The collision kernel is of BGK type with a general local Gibbs state. Assuming that the flow velocity is of the order of the Knudsen number, a Hilbert expansion yields a macroscopic model with finite temperature variations, whose complexity lies in between the hydrodynamic and the energy-transport equations. Its mathematical structure is explored and macroscopic models for specific examples of the global Gibbs state are presented.
Flame structure from a kinetic model for chemical reactions
Marzia Bisi, Maria Groppi and Giampiero Spiga
2010, 3(1): 17-34 doi: 10.3934/krm.2010.3.17 +[Abstract](289) +[PDF](243.9KB)
Abstract:
Steady one-dimensional flame structure is investigated in a binary mixture made up by two components of the same chemical species undergoing binary irreversible exothermic reactive encounters. A kinetic model at the Boltzmann level, accounting for chemical transitions as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of slow chemical reactions and collision dominated regime, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive Navier-Stokes equations at the fluid-dynamic scale. The resulting set of ordinary differential equations is investigated in the frame of the qualitative theory of dynamical systems, and numerical results are presented and briefly commented on for illustrative purposes.
Discrete velocity models of the Boltzmann equation and conservation laws
Alexander Bobylev, Mirela Vinerean and Åsa Windfäll
2010, 3(1): 35-58 doi: 10.3934/krm.2010.3.35 +[Abstract](492) +[PDF](336.1KB)
Abstract:
We consider in this paper the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVMs) of the classical (elastic) Boltzmann equation. We explain in detail how this problem can be solved and present a complete classification of (i.e. we present all distinct) normal plane DVMs with relatively small number $n$ of velocities ($n\leq 10$). Some results for models with larger number of velocities are also presented.
A mixed finite element method for nonlinear diffusion equations
Martin Burger, José A. Carrillo and Marie-Therese Wolfram
2010, 3(1): 59-83 doi: 10.3934/krm.2010.3.59 +[Abstract](561) +[PDF](4049.3KB)
Abstract:
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.
Entropy and chaos in the Kac model
Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss and Cédric Villani
2010, 3(1): 85-122 doi: 10.3934/krm.2010.3.85 +[Abstract](491) +[PDF](397.8KB)
Abstract:
We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.
A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation
Fausto Cavalli and Giovanni Naldi
2010, 3(1): 123-142 doi: 10.3934/krm.2010.3.123 +[Abstract](296) +[PDF](348.6KB)
Abstract:
In this work we introduce a new numerical approach for solving Cahn-Hilliard equation with Neumann boundary conditions involving recent mass transportation methods. The numerical scheme is based on an alternative formulation of the problem using the so called pseudo-inverse of the cumulative distribution function. We establish a stable fully discrete scheme that inherits the energy dissipation and mass conservation from the associated continuous problem. We perform some numerical experiments which confirm our results.
A model of sympatric speciation through reinforcement
Johan Henriksson, Torbjörn Lundh and Bernt Wennberg
2010, 3(1): 143-163 doi: 10.3934/krm.2010.3.143 +[Abstract](242) +[PDF](595.0KB)
Abstract:
Sympatric speciation, i.e. the evolutionary split of one species into two in the same environment, has been a highly troublesome concept. It has been a questioned if it is actually possible. Even though there have been a number of reported results both in the wild and from controlled experiments in laboratories, those findings are both hard to get and hard to analyze, or even repeat. In the current study we propose a mathematical model which addresses the question of sympatric speciation and the evolution of reinforcement. Our aim has been to capture some of the essential features such as: phenotype, resources, competition, heritage, mutation, and reinforcement, in as simple a way as possible. Still, the resulting model is not too easy to grasp with purely analytical tools, so we have also complemented those studies with stochastic simulations. We present a few results that both illustrates the usefulness of such a model, but also rises new biological questions about sympatric speciation and reinforcement in particular.
Fokker-Planck asymptotics for traffic flow models
Michael Herty and Lorenzo Pareschi
2010, 3(1): 165-179 doi: 10.3934/krm.2010.3.165 +[Abstract](406) +[PDF](1361.1KB)
Abstract:
Starting from microscopic interaction rules we derive kinetic models of Fokker-Planck type for vehicular traffic flow. The derivation is based on taking a suitable asymptotic limit of the corresponding Boltzmann model. As particular cases, the derived models comprise existing models. New Fokker-Planck models are also given and their differences to existing models are highlighted. Finally, we report on numerical experiments.
Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics
Zhongyi Huang, Peter A. Markowich and Christof Sparber
2010, 3(1): 181-194 doi: 10.3934/krm.2010.3.181 +[Abstract](309) +[PDF](1787.1KB)
Abstract:
We numerically study the three dimensional Gross-Pitaevskii equation for dipolar quantum gases using a time-splitting algorithm. We are mainly concerned with numerical investigations of the possible blow-up of solutions, i.e. collapse of the condensate, and the dynamics of vortices.

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