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KRM publishes high quality papers of original research in the areas of kinetic equations spanning from mathematical theory to numerical analysis, simulations and modelling. It includes studies on models arising from physics, engineering, finance, biology, human and social sciences, together with their related fields such as fluid models, interacting particle systems and quantum systems. A more detailed indication of its scope is given by the subject interests of the members of the Board of Editors. Invited expository articles are also published from time to time.
KRM was launched in 2008 and is edited by a group of energetic leaders to guarantee the journal's highest standard and closest link to the scientific communities. A unique feature of this journal is its streamlined review process and rapid publication. Authors are kept informed throughout the process through direct and personal communication between the authors and editors.
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 Publishes 4 issues a year in March, June, September and December.
 Publishes online only.
 Indexed in Science Citation Index, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
 Archived in Portico and CLOCKSS.
 KRM is a publication of the American Institute of Mathematical Sciences. All rights reserved.

TOP 10 Most Read Articles in KRM, November 2017
1 
On the dynamics of social conflicts: Looking for the black swan
Volume 6, Number 3, Pages: 459  479, 2013
Nicola Bellomo,
Miguel A. Herrero
and Andrea Tosin
Abstract
References
Full Text
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This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a Government, may give rise to strongly selfenhanced effects. The latter may be thought of as the early stages of massive unpredictable events known as Black Swans, although no analysis of any fullydeveloped Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to gametheoretical principles.

2 
Existence and sharp localization in velocity
of smallamplitude Boltzmann shocks
Volume 2, Number 4, Pages: 667  705, 2009
Guy Métivier
and K. Zumbrun
Abstract
Full Text
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Using a weighted $H^s$contraction mapping argument based on the
macromicro decomposition of Liu and Yu, we give an elementary proof
of existence, with sharp rates of decay and distance from the
ChapmanEnskog approximation, of smallamplitude shock profiles of
the Boltzmann equation with hardsphere potential, recovering and
slightly sharpening results obtained by Caflisch and Nicolaenko
using different techniques. A key technical point in both analyses
is that the linearized collision operator $L$ is negative definite
on its range, not only in the standard squareroot Maxwellian
weighted norm for which it is selfadjoint, but also in norms with
nearby weights. Exploring this issue further, we show that $L$ is
negative definite on its range in a much wider class of norms
including norms with weights asymptotic nearly to a full Maxwellian
rather than its square root. This yields sharp localization in
velocity at nearMaxwellian rate, rather than the squareroot rate
obtained in previous analyses.

3 
Global existence of weak solution to the free boundary problem for compressible NavierStokes
Volume 9, Number 1, Pages: 75  103, 2015
Zhenhua Guo
and Zilai Li
Abstract
References
Full Text
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In this paper, the compressible NavierStokes system (CNS) with constant viscosity coefficients is considered in three space dimensions. we prove the global existence of spherically symmetric weak solutions to the free boundary
problem for the CNS with vacuum and free boundary separating fluids and vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time.

4 
Kinetic derivation of fractional Stokes and StokesFourier systems
Volume 9, Number 1, Pages: 105  129, 2015
Sabine Hittmeir
and Sara MerinoAceituno
Abstract
References
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In recent works it has been demonstrated that using an appropriate rescaling, linear Boltzmanntype equations give rise to a scalar fractional diffusion equation in the limit of a small mean free path. The equilibrium distributions are typically heavytailed distributions, but also classical Gaussian equilibrium distributions allow for this phenomena if combined with a degenerate collision frequency for small velocities. This work aims to an extension in the sense that a linear BGKtype equation conserving not only mass, but also momentum and energy, for both mentioned regimes of equilibrium distributions is considered. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavytailed equilibria.

5 
Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules
Volume 9, Number 1, Pages: 1  49, 2015
Kleber Carrapatoso
Abstract
References
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We prove a quantitative propagation of chaos and entropic chaos, uniformly in time, for the spatially homogeneous Landau equation in the case of Maxwellian molecules. We improve the results of Fontbona, Guérin and Méléard [9] and Fournier [10] where the propagation of chaos is proved for finite time. Moreover, we prove a quantitative estimate on the rate of convergence to equilibrium uniformly in the number of particles.

6 
Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction
Volume 9, Number 1, Pages: 165  191, 2015
Xueke Pu
and Boling Guo
Abstract
References
Full Text
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The hydrodynamic equations with quantum effects are studied in this paper. First we establish the global existence of smooth solutions with small initial data and then in the second part, we establish the convergence of the solutions of the quantum hydrodynamic equations to those of the classical hydrodynamic equations. The energy equation is considered in this paper, which added new difficulties to the energy estimates, especially to the selection of the appropriate Sobolev spaces.

7 
Asymptotic preserving scheme for a kinetic model describing incompressible fluids
Volume 9, Number 1, Pages: 51  74, 2015
Nicolas Crouseilles,
Mohammed Lemou,
SV Raghurama Rao,
Ankit Ruhi
and Muddu Sekhar
Abstract
References
Full Text
Related Articles
The kinetic theory of fluid turbulence modeling developed by Degond and Lemou in [7]
is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for
turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multidimensional prototype model and first propose an appropriate change of frame that
makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term
is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a onedimensional
framework on the basis of the developed strategy to confirm its efficiency.

8 
Global existence and steady states of a two competing species KellerSegel chemotaxis model
Volume 8, Number 4, Pages: 777  807, 2015
Qi Wang,
Lu Zhang,
Jingyue Yang
and Jia Hu
Abstract
References
Full Text
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We study a onedimensional quasilinear system proposed by J. Tello and M. Winkler [27] which models the population dynamics of two competing species attracted by the same chemical. The kinetic terms of the interacting species are chosen to be of the LotkaVolterra type and the boundary conditions are of homogeneous Neumann type which represent an enclosed domain. We prove the global existence and boundedness of classical solutions to the fully parabolic system. Then we establish the existence of nonconstant positive steady states through bifurcation theory. The stability or instability of the bifurcating solutions is investigated rigorously. Our results indicate that small intervals support stable monotone positive steady states and large intervals support nonmonotone steady states. Finally, we perform extensive numerical studies to demonstrate and verify our theoretical results. Our numerical simulations also illustrate the formation of stable steady states and timeperiodic solutions with various interesting spatial structures.

9 
A kinetic model for the formation of swarms with nonlinear interactions
Volume 9, Number 1, Pages: 131  164, 2015
Martin Parisot
and Mirosław Lachowicz
Abstract
References
Full Text
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The present paper deals with the modeling of formation and destruction of swarms using a nonlinear Boltzmannlike
equation. We introduce a new model that contains parameters characterizing the attractiveness or repulsiveness of
individuals. The model can represent both gregarious and solitarious behaviors.
In the latter case we provide a mathematical analysis in the space homogeneous case. Moreover we identify relevant
hydrodynamic limits on a formal way. We introduce some preliminary results in the case of gregarious behavior and
we indicate open problems for further research.
Finally, we provide numerical simulations to illustrate the ability of the model to represent formation or destruction
of swarms.

10 
On the weak coupling limit of quantum manybody dynamics and the quantum Boltzmann equation
Volume 8, Number 3, Pages: 443  465, 2015
Xuwen Chen
and Yan Guo
Abstract
References
Full Text
Related Articles
The rigorous derivation of the UehlingUhlenbeck equation from more
fundamental quantum manyparticle systems is a challenging open problem in
mathematics. In this paper, we exam the weak coupling limit of quantum $N$
particle dynamics. We assume the integral of the microscopic interaction is
zero and we assume $W^{4,1}$ perparticle regularity on the coressponding
BBGKY sequence so that we can rigorously commute limits and integrals. We
prove that, if the BBGKY sequence does converge in some weak sense, then
this weakcoupling limit must satisfy the infinite quantum MaxwellBoltzmann
hierarchy instead of the expected infinite UehlingUhlenbeck hierarchy,
regardless of the statistics the particles obey. Our result indicates that,
in order to derive the UehlingUhlenbeck equation, one must work with
perparticle regularity bound below $W^{4,1}$.

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