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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.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 6 issues a year in February, April, June, August, October 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.

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Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D
Maxime Hauray and Samir Salem
2019, 12(2) : 269-302 doi: 10.3934/krm.2019012 +[Abstract](110) +[HTML](65) +[PDF](617.67KB)
Abstract:

We consider a particle system in 1D, interacting via repulsive or attractive Coulomb forces. We prove the trajectorial propagation of molecular chaos towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck equation. We obtain a quantitative estimate of convergence in the mean in MKW metric of order one, with an optimal convergence rate of order $N^{-1/2}$. We also prove some exponential concentration inequalities of the associated empirical measures. A key argument is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we are able to adapt for the particle system in some sense.

Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion
Marco Di Francesco and Yahya Jaafra
2019, 12(2) : 303-322 doi: 10.3934/krm.2019013 +[Abstract](108) +[HTML](58) +[PDF](700.09KB)
Abstract:

In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

On a Boltzmann equation for Haldane statistics
Leif Arkeryd and Anne Nouri
2019, 12(2) : 323-346 doi: 10.3934/krm.2019014 +[Abstract](101) +[HTML](63) +[PDF](446.56KB)
Abstract:

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper Haldane [10] proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong $L^1$ solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stabililty. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system
Peter Markowich and Jesús Sierra
2019, 12(2) : 347-356 doi: 10.3934/krm.2019015 +[Abstract](80) +[HTML](63) +[PDF](352.77KB)
Abstract:

We investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization.

A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data
Bingkang Huang and Lan Zhang
2019, 12(2) : 357-396 doi: 10.3934/krm.2019016 +[Abstract](105) +[HTML](78) +[PDF](561.64KB)
Abstract:

We present a two-dimensional coupled system for particles and compressible conducting fluid in an electromagnetic field interactions, which the kinetic Vlasov-Fokker-Planck model for particle part and the isentropic compressible MHD equations for the fluid part, respectively, and these separate systems are coupled with the drag force. For this specific coupled system, a sufficient framework for the global existence of classical solutions with large initial data which may contain vacuum is established.

On the interplay between behavioral dynamics and social interactions in human crowds
Nicola Bellomo, Livio Gibelli and Nisrine Outada
2019, 12(2) : 397-409 doi: 10.3934/krm.2019017 +[Abstract](97) +[HTML](55) +[PDF](656.01KB)
Abstract:

This paper presents a computational modeling approach to the dynamics of human crowds, where social interactions can have an important influence on the behavioral dynamics of pedestrians. The modeling of the contagion and propagation of emotional states is carried out by looking at real physical situations where safety problems might arise in some specific circumstances. The approach is based on the methods of the kinetic theory of active particles. The evacuation of a metro station is simulated to enlighten the role of the emotional state in the overall dynamics.

Emergence of aggregation in the swarm sphere model with adaptive coupling laws
Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee and Se Eun Noh
2019, 12(2) : 411-444 doi: 10.3934/krm.2019018 +[Abstract](96) +[HTML](78) +[PDF](561.94KB)
Abstract:

We present aggregation estimates for the swarm sphere model equipped with the adaptive coupling laws on a sphere. The temporal evolution of coupling strength is determined by a feedback rule incorporating the balance between relative spatial variations and linear damping. For the analytical treatment, we employ two adaptive feedback laws, namely anti-Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient framework leading to the complete aggregation in which all particles aggregate to the same position and behave like one big point cluster asymptotically. Our frameworks are given in terms of the initial positions and the coupling strengths. For the Hebbian law, we provide proper subsets of the basin of attractions for the complete aggregation and bi-polar aggregation where particles aggregate to the north pole and south pole simultaneously. We also present a uniform \begin{document}$\ell_p$\end{document}-stability of the swarm sphere model with an adaptive coupling with respect to the initial data when the complete aggregation occurs exponentially fast.

Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness
Marcel Braukhoff
2019, 12(2) : 445-482 doi: 10.3934/krm.2019019 +[Abstract](87) +[HTML](57) +[PDF](575.17KB)
Abstract:

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice:

This system contains an interaction potential \begin{document}$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$\end{document} being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, \begin{document}$ε(p) = -\sum_{i = 1}^d$\end{document} \begin{document}$\cos(2π p_i)$\end{document} is the dispersion relation and \begin{document}$\mathcal{F}_f$\end{document} denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on \begin{document}$f$\end{document} in this context.

In a dilute plasma—without collisions (r.h.s\begin{document}$. = 0$\end{document})—this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.

From particle to kinetic and hydrodynamic descriptions of flocking
Seung-Yeal Ha and Eitan Tadmor
2008, 1(3) : 415-435 doi: 10.3934/krm.2008.1.415 +[Abstract](1265) +[PDF](271.2KB) Cited By(141)
Mathematical theory and numerical methods for Bose-Einstein condensation
Weizhu Bao and Yongyong Cai
2013, 6(1) : 1-135 doi: 10.3934/krm.2013.6.1 +[Abstract](1443) +[PDF](3152.1KB) Cited By(104)
Double milling in self-propelled swarms from kinetic theory
José A. Carrillo, M. R. D’Orsogna and V. Panferov
2009, 2(2) : 363-378 doi: 10.3934/krm.2009.2.363 +[Abstract](1285) +[PDF](299.0KB) Cited By(103)
Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
Marion Acheritogaray, Pierre Degond, Amic Frouvelle and Jian-Guo Liu
2011, 4(4) : 901-918 doi: 10.3934/krm.2011.4.901 +[Abstract](1257) +[PDF](409.3KB) Cited By(67)
On the dynamics of social conflicts: Looking for the black swan
Nicola Bellomo, Miguel A. Herrero and Andrea Tosin
2013, 6(3) : 459-479 doi: 10.3934/krm.2013.6.459 +[Abstract](951) +[PDF](702.8KB) Cited By(41)
Towards a mathematical theory of complex socio-economical systems by functional subsystems representation
Giulia Ajmone Marsan, Nicola Bellomo and Massimo Egidi
2008, 1(2) : 249-278 doi: 10.3934/krm.2008.1.249 +[Abstract](785) +[PDF](329.0KB) Cited By(40)
The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients
Quansen Jiu and Zhouping Xin
2008, 1(2) : 313-330 doi: 10.3934/krm.2008.1.313 +[Abstract](1042) +[PDF](247.8KB) Cited By(36)
On a chemotaxis model with saturated chemotactic flux
Alina Chertock, Alexander Kurganov, Xuefeng Wang and Yaping Wu
2012, 5(1) : 51-95 doi: 10.3934/krm.2012.5.51 +[Abstract](1118) +[PDF](1412.8KB) Cited By(36)
Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff
Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai and Tong Yang
2008, 1(3) : 453-489 doi: 10.3934/krm.2008.1.453 +[Abstract](1065) +[PDF](398.4KB) Cited By(29)
Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity
Feimin Huang, Yi Wang and Tong Yang
2010, 3(4) : 685-728 doi: 10.3934/krm.2010.3.685 +[Abstract](1029) +[PDF](606.8KB) Cited By(28)
Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks
Alin Pogan and Kevin Zumbrun
2019, 12(1) : 1-36 doi: 10.3934/krm.2019001 +[Abstract](484) +[HTML](193) +[PDF](632.1KB) PDF Downloads(91)
Letter to the editors in chief
Tai-Ping Liu and Shih-Hsien Yu
2018, 11(1) : 215-217 doi: 10.3934/krm.2018011 +[Abstract](2362) +[HTML](286) +[PDF](273.0KB) PDF Downloads(78)
Entropy production inequalities for the Kac Walk
Eric A. Carlen, Maria C. Carvalho and Amit Einav
2018, 11(2) : 219-238 doi: 10.3934/krm.2018012 +[Abstract](902) +[HTML](284) +[PDF](477.57KB) PDF Downloads(72)
A Vlasov-Poisson plasma of infinite mass with a point charge
Gang Li and Xianwen Zhang
2018, 11(2) : 303-336 doi: 10.3934/krm.2018015 +[Abstract](882) +[HTML](286) +[PDF](511.26KB) PDF Downloads(70)
Linear Boltzmann equation and fractional diffusion
Claude Bardos, François Golse and Ivan Moyano
2018, 11(4) : 1011-1036 doi: 10.3934/krm.2018039 +[Abstract](647) +[HTML](150) +[PDF](470.71KB) PDF Downloads(69)
A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures
Clément Jourdana and Paola Pietra
2019, 12(1) : 217-242 doi: 10.3934/krm.2019010 +[Abstract](558) +[HTML](197) +[PDF](1843.94KB) PDF Downloads(64)
Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density
Fei Chen, Boling Guo and Xiaoping Zhai
2019, 12(1) : 37-58 doi: 10.3934/krm.2019002 +[Abstract](482) +[HTML](175) +[PDF](487.82KB) PDF Downloads(63)
A general consistent BGK model for gas mixtures
Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga and Irina F. Potapenko
2018, 11(6) : 1377-1393 doi: 10.3934/krm.2018054 +[Abstract](558) +[HTML](97) +[PDF](459.44KB) PDF Downloads(63)
Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary
Linjie Xiong
2018, 11(3) : 469-490 doi: 10.3934/krm.2018021 +[Abstract](703) +[HTML](207) +[PDF](497.17KB) PDF Downloads(62)
Time-splitting methods to solve the Hall-MHD systems with Lévy noises
Zhong Tan, Huaqiao Wang and Yucong Wang
2019, 12(1) : 243-267 doi: 10.3934/krm.2019011 +[Abstract](430) +[HTML](161) +[PDF](491.35KB) PDF Downloads(60)

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