# American Institue of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

## Journal Home

All Issues

### Volume 1, 2008

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.

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

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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)
Abstract:

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.

Tomasz Komorowski  and  Łukasz Stȩpień
2018, 11(2) : 239-278 doi: 10.3934/krm.2018013 +[Abstract](14) +[HTML](17) +[PDF](615.36KB)
Abstract:

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.

2018, 11(2) : 279-301 doi: 10.3934/krm.2018014 +[Abstract](19) +[HTML](13) +[PDF](473.56KB)
Abstract:

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.

Gang Li  and  Xianwen Zhang
2018, 11(2) : 303-336 doi: 10.3934/krm.2018015 +[Abstract](23) +[HTML](12) +[PDF](511.26KB)
Abstract:

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.

Marco Torregrossa  and  Giuseppe Toscani
2018, 11(2) : 337-355 doi: 10.3934/krm.2018016 +[Abstract](27) +[HTML](14) +[PDF](477.51KB)
Abstract:

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.

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)
Abstract:

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.

2018, 11(2) : 397-408 doi: 10.3934/krm.2018018 +[Abstract](13) +[HTML](13) +[PDF](366.63KB)
Abstract:

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.

2018, 11(2) : 409-439 doi: 10.3934/krm.2018019 +[Abstract](18) +[HTML](17) +[PDF](718.78KB)
Abstract:

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.

2008, 1(3) : 415-435 doi: 10.3934/krm.2008.1.415 +[Abstract](89) +[PDF](271.2KB) Cited By(131)
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)
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)
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)
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)
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)
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)
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)
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)
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
Caochuan Ma  , Zaihong Jiang  and  Renhui Wan
Victor A. Kovtunenko  and  Anna V. Zubkova
Gang Li  and  Xianwen Zhang
Jishan Fan  and  Yueling Jia
Fabrice Baudoin  and  Camille Tardif
Shuguang Shao  , Shu Wang  and  Wen-Qing Xu

2016  Impact Factor: 1.261