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Volume 11, 2018

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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.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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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](17) +[HTML](33) +[PDF](477.57KB)

Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of 'chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the 'Almost' Cercignani Conjecture on the sphere.

Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation
Tomasz Komorowski  and  Łukasz Stȩpień 
2018, 11(2) : 239-278 doi: 10.3934/krm.2018013 +[Abstract](14) +[HTML](17) +[PDF](615.36KB)

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion
Maxime Breden 
2018, 11(2) : 279-301 doi: 10.3934/krm.2018014 +[Abstract](19) +[HTML](13) +[PDF](473.56KB)

In this paper, we investigate the use of so called "duality lemmas" to study the system of discrete coagulation-fragmentation equations with diffusion. When the fragmentation is strong enough with respect to the coagulation, we show that we have creation and propagation of superlinear moments. In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [3], about the regularity of the solutions in the pure coagulation case, to strong fragmentation models.

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](23) +[HTML](12) +[PDF](511.26KB)

We study the time evolution of the three dimensional Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass. We prove the existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space, but not integrable. This result extends a previous theorem for Yukawa potential obtained in [10] to the case of Coulomb interaction.

On a Fokker-Planck equation for wealth distribution
Marco Torregrossa  and  Giuseppe Toscani 
2018, 11(2) : 337-355 doi: 10.3934/krm.2018016 +[Abstract](27) +[HTML](14) +[PDF](477.51KB)

We study here a Fokker-Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society [2, 10, 22]. In particular, we analyze the large-time behavior of the solution, by showing that convergence to the steady state can be obtained in various norms at different rates.

Invariant measures for a stochastic Fokker-Planck equation
Sylvain De Moor  , Luis Miguel Rodrigues  and  Julien Vovelle 
2018, 11(2) : 357-395 doi: 10.3934/krm.2018017 +[Abstract](15) +[HTML](16) +[PDF](641.91KB)

We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term. When the noise intensity is not too large, we solve the corresponding Cauchy problem in a space of functions ensuring good localization in the velocity variable. Then we show under similar conditions that the generated dynamics, with prescribed total mass, admits a unique invariant measure which is exponentially mixing. The proof relies on hypocoercive estimates and hypoelliptic regularity. At last we provide an explicit example showing that our analytic framework does require some smallness condition on the noise intensity.

Regularity theorems for a biological network formulation model in two space dimensions
Xiangsheng Xu 
2018, 11(2) : 397-408 doi: 10.3934/krm.2018018 +[Abstract](13) +[HTML](13) +[PDF](366.63KB)

We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [13]. A consequence of these results is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.

Numerical schemes for kinetic equation with diffusion limit and anomalous time scale
Hivert Hélène 
2018, 11(2) : 409-439 doi: 10.3934/krm.2018019 +[Abstract](18) +[HTML](17) +[PDF](718.78KB)

In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form \begin{document}${\varepsilon ^2}\left( {1 + \left| {\ln \left( \varepsilon \right)} \right|} \right)$\end{document}. When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. This work deals with a critical exponent between these two cases, for which an anomalous time scale must be used to find a standard diffusion limit. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degeneracy \begin{document}$ \varepsilon\to0$\end{document} makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Moreover, we prove that the convergence towards the diffusion limit happens with two scales, the second being very slow. The schemes we propose are designed to respect this asymptotic behavior. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

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](89) +[PDF](271.2KB) Cited By(131)
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](86) +[PDF](3152.1KB) Cited By(98)
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](83) +[PDF](299.0KB) Cited By(97)
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](82) +[PDF](409.3KB) Cited By(59)
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](83) +[PDF](702.8KB) Cited By(38)
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](59) +[PDF](329.0KB) Cited By(37)
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](126) +[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](48) +[PDF](1412.8KB) Cited By(35)
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](185) +[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](75) +[PDF](606.8KB) Cited By(26)
Tai-Ping Liu  and  Shih-Hsien Yu 
2018, 11(1) : 215-217 doi: 10.3934/krm.2018011 +[Abstract](1251) +[HTML](6) +[PDF](273.0KB) PDF Downloads(11)
Local well-posedness for the tropical climate model with fractional velocity diffusion
Caochuan Ma  , Zaihong Jiang  and  Renhui Wan 
2016, 9(3) : 551-570 doi: 10.3934/krm.2016006 +[Abstract](68) +[PDF](443.2KB) PDF Downloads(7)
Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
Victor A. Kovtunenko  and  Anna V. Zubkova 
2018, 11(1) : 119-135 doi: 10.3934/krm.2018007 +[Abstract](853) +[HTML](33) +[PDF](452.3KB) PDF Downloads(5)
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](23) +[HTML](12) +[PDF](511.26KB) PDF Downloads(5)
Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum
Jishan Fan  and  Yueling Jia 
2018, 11(1) : 97-106 doi: 10.3934/krm.2018005 +[Abstract](732) +[HTML](2) +[PDF](324.7KB) PDF Downloads(5)
Hypocoercive estimates on foliations and velocity spherical Brownian motion
Fabrice Baudoin  and  Camille Tardif 
2018, 11(1) : 1-23 doi: 10.3934/krm.2018001 +[Abstract](737) +[HTML](5) +[PDF](417.3KB) PDF Downloads(5)
On half-space problems for the weakly non-linear discrete Boltzmann equation
Niclas Bernhoff 
2010, 3(2) : 195-222 doi: 10.3934/krm.2010.3.195 +[Abstract](55) +[PDF](331.5KB) PDF Downloads(4)
Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
Shuguang Shao  , Shu Wang  and  Wen-Qing Xu 
2018, 11(1) : 179-190 doi: 10.3934/krm.2018009 +[Abstract](736) +[HTML](5) +[PDF](376.0KB) PDF Downloads(4)
Compressible Euler equations interacting with incompressible flow
Young-Pil Choi 
2015, 8(2) : 335-358 doi: 10.3934/krm.2015.8.335 +[Abstract](44) +[PDF](479.2KB) PDF Downloads(3)
$L^\infty$ resolvent bounds for steady Boltzmann's equation
Kevin Zumbrun 
2017, 10(4) : 1255-1257 doi: 10.3934/krm.2017048 +[Abstract](44) +[HTML](2) +[PDF](304.2KB) PDF Downloads(3)

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