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Volume 1, 2008

Kinetic & Related Models

2008 , Volume 1 , Issue 1

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Pierre Degond , Seiji Ukai and  Tong Yang
2008, 1(1): i-ii doi: 10.3934/krm.2008.1.1i +[Abstract](40) +[PDF](26.2KB)
Kinetic theory is probably one of the most efficient and important theories allowing to bridge the microscopic and macroscopic descriptions of a variety of dynamical phenomena in many fields of science, technology, and more generally, in virtually all domains of knowledge. Originally rooted in the theory of rarefied gases since the seminal works of Boltzmann and Maxwell in the 19th century, followed by landmarks established by Hilbert, Chapman and Enskog, Carleman, Grad, and more recent mathematicians, kinetic theory has expanded to many new areas of applications, ranging from physics to economics and social sciences including especially modern fields such as biology, epidemiology, and genetics.

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Analysis of a model for wealth redistribution
Daniel Matthes and  Giuseppe Toscani
2008, 1(1): 1-27 doi: 10.3934/krm.2008.1.1 +[Abstract](37) +[PDF](317.2KB)
A recent application of the kinetic theory for many particle systems is the description of the redistribution of wealth among trading agents in a simple market economy. This paper provides an analytical investigation of the particular model with quenched saving propensities, which has been introduced by Chakrabarti, Chatterjee and Manna [11]. We prove uniqueness and dynamical stability of the stationary solution to the underlying Boltzmann equation, and provide estimates on the rate of equilibration. As one main result, we obtain that realistic steady wealth distributions with Pareto tail are only algebraically stable in this framework.
Global existence for the kinetic chemotaxis model without pointwise memory effects, and including internal variables
Nikolaos Bournaveas and  Vincent Calvez
2008, 1(1): 29-48 doi: 10.3934/krm.2008.1.29 +[Abstract](48) +[PDF](268.7KB)
This paper is concerned with the kinetic model of Othmer-Dunbar-Alt for bacterial motion. Following a previous work, we apply the dispersion and Strichartz estimates to prove global existence under several borderline growth assumptions on the turning kernel. In particular we study the kinetic model with internal variables taking into account the complex molecular network inside the cell.
Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space
Yoshihiro Ueda , Tohru Nakamura and  Shuichi Kawashima
2008, 1(1): 49-64 doi: 10.3934/krm.2008.1.49 +[Abstract](61) +[PDF](230.3KB)
In this paper, we consider the large-time behavior of solutions to the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space. We show that the solution to the problem converges to the corresponding planar stationary wave as time tends to infinity under smallness condition on the initial perturbation. It is proved that the tangential derivatives of the solution verify quantitative decay estimates for $t\to\infty$. Moreover, an additional algebraic convergence rate is obtained by assuming that the initial perturbation decays algebraically in the normal direction. The crucial point of the proof is to derive a priori estimates of solutions by using the time and space weighted energy method.
A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface
Shi Jin , Xu Yang and  Guangwei Yuan
2008, 1(1): 65-84 doi: 10.3934/krm.2008.1.65 +[Abstract](33) +[PDF](404.4KB)
When a linear transport equation contains two scales, one diffusive and the other non-diffusive, it is natural to use a domain decomposition method which couples the transport equation with a diffusion equation with an interface condition. One such method was introduced by Golse, Jin and Levermore in [11], where an interface condition, which is derived from the conservation of energy density, was used to construct an efficient non-iterative domain decomposition method.
    In this paper, we extend this domain decomposition method to diffusive interfaces where the energy flux is conserved. Such problems arise in high frequency waves in random media. New operators corresponding to transmission and reflections at the interfaces are derived and then used in the interface conditions. With these new operators we are able to construct both first and second order (in terms of the mean free path) non-iterative domain decomposition methods of the type by Golse-Jin-Levermore, which will be proved having the desired accuracy and tested numerically.
Self-averaging of kinetic models for waves in random media
Guillaume Bal and  Olivier Pinaud
2008, 1(1): 85-100 doi: 10.3934/krm.2008.1.85 +[Abstract](45) +[PDF](231.2KB)
Kinetic equations are often appropriate to model the energy density of high frequency waves propagating in highly heterogeneous media. The limitations of the kinetic model are quantified by the statistical instability of the wave energy density, i.e., by its sensitivity to changes in the realization of the underlying heterogeneous medium modeled as a random medium. In the simplified Itô-Schrödinger regime of wave propagation, we obtain optimal estimates for the statistical instability of the wave energy density for different configurations of the source terms and the domains over which the energy density is measured. We show that the energy density is asymptotically statistically stable (self-averaging) in many configurations. In the case of highly localized source terms, we obtain an explicit asymptotic expression for the scintillation function in the high frequency limit.
Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme
Maike Schulte and  Anton Arnold
2008, 1(1): 101-125 doi: 10.3934/krm.2008.1.101 +[Abstract](77) +[PDF](4086.8KB)
We consider the two-dimensional time-dependent Schrödinger equation with the new compact nine-point scheme in space and the Crank-Nicolson difference scheme in time. For the resulting difference equation we derive discrete transparent boundary conditions in order to get highly accurate solutions for open boundary problems. Numerical experiments illustrate the perfect absorption of outgoing wave independently of their impact angle at the boundary. Finally, we apply inhomogeneous discrete transparent boundary conditions to the transient simulation of quantum waveguides.
Fast-reaction limit for the inhomogeneous Aizenman-Bak model
José A. Carrillo , Laurent Desvillettes and  Klemens Fellner
2008, 1(1): 127-137 doi: 10.3934/krm.2008.1.127 +[Abstract](44) +[PDF](172.4KB)
Solutions of the spatially inhomogeneous diffusive\linebreak Aizenmann-Bak model for clustering within a bounded domain with homogeneous Neumann boundary conditions are shown to stabilize, in the fast reaction limit, towards local equilibria determined by their monomer density. Moreover, the sequence of monomer densities converges to the solution of a nonlinear diffusion equation whose nonlinearity depends on the size-dependent diffusion coefficient. Initial data are assumed to be integrable, bounded and with a certain number of moments in size. The number density of clusters for the solutions is assumed to verify uniform bounds away from zero and infinity independently of the scale parameter.
Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case
Mihai Bostan and  Thierry Goudon
2008, 1(1): 139-170 doi: 10.3934/krm.2008.1.139 +[Abstract](38) +[PDF](339.3KB)
We study the asymptotic regime for the relativistic Vlasov-Maxwell-Fokker-Planck system which corresponds to a mean free path small compared to the Debye length, chosen as an observation length scale, combined to a large thermal velocity assumption. We are led to a convection-diffusion equation, where the convection velocity is obtained by solving a Poisson equation. The analysis is performed in the one and one half dimensional case and the proof combines dissipation mechanisms and finite speed of propagation properties.

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