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Kinetic & Related Models

2012 , Volume 5 , Issue 4

Issue on Variational Methods in
Nonlinear Elliptic Equations

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Exponential stability of the solutions to the Boltzmann equation for the Benard problem
Leif Arkeryd , Raffaele Esposito , Rossana Marra and  Anne Nouri
2012, 5(4): 673-695 doi: 10.3934/krm.2012.5.673 +[Abstract](40) +[PDF](475.5KB)
Abstract:
We complete the result in [2] by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.
A model for the evolution of traffic jams in multi-lane
Florent Berthelin and  Damien Broizat
2012, 5(4): 697-728 doi: 10.3934/krm.2012.5.697 +[Abstract](31) +[PDF](1113.8KB)
Abstract:
In [8], Berthelin, Degond, Delitala and Rascle introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model consists of a Pressureless Gas Dynamics system under a maximal constraint on the density and is derived through a singular limit of the Aw-Rascle model. In the present paper we propose an improvement of this model by allowing the road to be multi-lane piecewise. The idea is to use the maximal constraint to model the number of lanes. We also add in the model a parameter $\alpha$ which model the various speed limitations according to the number of lanes. We present the dynamical behaviour of clusters (traffic jams) and by approximation with such solutions, we obtain an existence result of weak solutions for any initial data.
Time evolution of a Vlasov-Poisson plasma with magnetic confinement
Silvia Caprino , Guido Cavallaro and  Carlo Marchioro
2012, 5(4): 729-742 doi: 10.3934/krm.2012.5.729 +[Abstract](38) +[PDF](356.2KB)
Abstract:
We study the time evolution of a Vlasov-Poisson plasma moving in an infinite cylinder, in which it is confined by an unbounded external magnetic field. This field depends only on the distance from the border of the cylinder, is tangent to the border and singular on it. We prove the existence and uniqueness of the solution, and also its confinement inside the cylinder for all times, i.e. the external field behaves like a magnetic mirror. Possible generalizations are discussed.
Global existence in critical spaces for the compressible magnetohydrodynamic equations
Qing Chen and  Zhong Tan
2012, 5(4): 743-767 doi: 10.3934/krm.2012.5.743 +[Abstract](52) +[PDF](466.8KB)
Abstract:
In this paper, we are concerned with the global existence and uniqueness of the strong solutions to the compressible Magnetohydrodynamic equations in $\mathbb{R}^N(N\ge3)$. Under the condition that the initial data are close to an equilibrium state with constant density, temperature and magnetic field, we prove the global existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations.
On the gain of regularity for the positive part of Boltzmann collision operator associated with soft-potentials
Yong-Kum Cho and  Hera Yun
2012, 5(4): 769-786 doi: 10.3934/krm.2012.5.769 +[Abstract](28) +[PDF](434.8KB)
Abstract:
As for the positive part of Boltzmann's collision operator associated with the collision kernel of soft-potential type, we evaluate its Fourier transform explicitly and prove a set of bilinear estimates for $L^p$ and Sobolev regularity.
Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles
Anaïs Crestetto , Nicolas Crouseilles and  Mohammed Lemou
2012, 5(4): 787-816 doi: 10.3934/krm.2012.5.787 +[Abstract](39) +[PDF](1100.3KB)
Abstract:
This work is devoted to the numerical simulation of the Vlasov equation in the fluid limit using particles. To that purpose, we first perform a micro-macro decomposition as in [3] where asymptotic preserving schemes have been derived in the fluid limit. In [3], a uniform grid was used to approximate both the micro and the macro part of the full distribution function. Here, we modify this approach by using a particle approximation for the kinetic (micro) part, the fluid (macro) part being always discretized by standard finite volume schemes. There are many advantages in doing so: $(i)$ the so-obtained scheme presents a much less level of noise compared to the standard particle method; $(ii)$ the computational cost of the micro-macro model is reduced in the fluid regime since a small number of particles is needed for the micro part; $(iii)$ the scheme is asymptotic preserving in the sense that it is consistent with the kinetic equation in the rarefied regime and it degenerates into a uniformly (with respect to the Knudsen number) consistent (and deterministic) approximation of the limiting equation in the fluid regime.
From individual to collective behaviour of coupled velocity jump processes: A locust example
Radek Erban and  Jan Haskovec
2012, 5(4): 817-842 doi: 10.3934/krm.2012.5.817 +[Abstract](48) +[PDF](574.3KB)
Abstract:
A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on the behaviour of locusts. It exhibits nontrivial dynamics with a pitchfork bifurcation and recovers the observed group directional switching. Estimates of the expected switching times, in terms of the number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations (with nonlocal and nonlinear right hand side) is derived and analyzed. The existence of its solutions is proven and the system's long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model.
Coupling of non-local driving behaviour with fundamental diagrams
Michael Herty and  Reinhard Illner
2012, 5(4): 843-855 doi: 10.3934/krm.2012.5.843 +[Abstract](32) +[PDF](8419.8KB)
Abstract:
We present an extended discussion of a macroscopic traffic flow model [18] which includes non-local and relaxation terms for vehicular traffic flow on unidirectional roads. The braking and acceleration forces are based on a behavioural model and on free flow dynamics. The latter are modelled by using different fundamental diagrams. Numerical investigations for different situations illustrate the properties of the mathematical model. In particular, the emergence of stop-and-go waves is observed for suitable parameter ranges.
Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities
Masashi Ohnawa
2012, 5(4): 857-872 doi: 10.3934/krm.2012.5.857 +[Abstract](31) +[PDF](429.6KB)
Abstract:
The present paper is concerned with the asymptotic behavior of a discontinuous solution to a model system of radiating gas. As we assume that an initial data has a discontinuity only at one point, so does the solution. Here the discontinuous solution is supposed to satisfy an entropy condition in the sense of Kruzkov. Previous researches have shown that the solution converges uniformly to a traveling wave if an initial perturbation is integrable and is small in the suitable Sobolev space. If its anti-derivative is also integrable, the convergence rate is known to be $(1+t)^{-1/4}$ as time $t$ tends to infinity. In the present paper, we improve the previous result and show that the convergence rate is exactly the same as the spatial decay rate of the initial perturbation.
An age-structured two-sex model in the space of radon measures: Well posedness
Agnieszka Ulikowska
2012, 5(4): 873-900 doi: 10.3934/krm.2012.5.873 +[Abstract](27) +[PDF](493.3KB)
Abstract:
In the following paper a well-posedness of an age-structured two-sex population model in a space of Radon measures equipped with a flat metric is presented. Existence and uniqueness of measure valued solutions is proved by a regularization technique. This approach allows to obtain Lipschitz continuity of solutions with respect to time and stability estimates. Moreover, a brief discussion on a marriage function, which is the main source of a nonlinearity, is carried out and an example of the marriage function fitting into this framework is given.

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