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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.
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  • Archived in Portico and CLOCKSS.
  • KRM is a publication of the American Institute of Mathematical Sciences. All rights reserved.

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A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff
Nicolas Fournier
2019, 12(3) : 483-505 doi: 10.3934/krm.2019020 +[Abstract](34) +[HTML](12) +[PDF](496.89KB)
Abstract:

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution \begin{document}$ (f_t)_{t\geq 0} $\end{document}, once the initial condition \begin{document}$ f_0 $\end{document} with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a \begin{document}$ (0,\infty)\times {\mathbb{R}}^3 $\end{document} random variable \begin{document}$ (M_t,V_t) $\end{document} such that \begin{document}$ \mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t $\end{document}. We also write down a series expansion of \begin{document}$ f_t $\end{document}. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express \begin{document}$ f_t $\end{document} in terms of \begin{document}$ f_0 $\end{document}, we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [18] and of its interpretation by McKean [10,11].

Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation
Torsten Keßler and Sergej Rjasanow
2019, 12(3) : 507-549 doi: 10.3934/krm.2019021 +[Abstract](42) +[HTML](12) +[PDF](1793.66KB)
Abstract:

In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
Étienne Bernard, Marie Doumic and Pierre Gabriel
2019, 12(3) : 551-571 doi: 10.3934/krm.2019022 +[Abstract](35) +[HTML](13) +[PDF](797.98KB)
Abstract:

We study the asymptotic behaviour of the following linear growth-fragmentation equation

and prove that under fairly general assumptions on the division rate \begin{document}$ B(x), $\end{document} its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypocoercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted \begin{document}$ L^2 $\end{document} space, where well-posedness is obtained via semigroup analysis. We also propose a non-diffusive numerical scheme, able to capture the oscillations.

Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition
Young-Pil Choi and Samir Salem
2019, 12(3) : 573-592 doi: 10.3934/krm.2019023 +[Abstract](35) +[HTML](10) +[PDF](414.53KB)
Abstract:

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocity-dependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.

Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations
Maxime Herda and Luis Miguel Rodrigues
2019, 12(3) : 593-636 doi: 10.3934/krm.2019024 +[Abstract](30) +[HTML](10) +[PDF](698.9KB)
Abstract:

We consider various sets of Vlasov-Fokker-Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them in a regime where the strength of the magnetic field is effectively stronger than that of collisions we first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached our asymptotic models capture a non trivial dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation
Teng Wang and Yi Wang
2019, 12(3) : 637-679 doi: 10.3934/krm.2019025 +[Abstract](31) +[HTML](17) +[PDF](692.44KB)
Abstract:

We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24,22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.

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](1611) +[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](1807) +[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](1656) +[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](1553) +[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](1225) +[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](1028) +[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](1321) +[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](1437) +[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](1336) +[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](1304) +[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](695) +[HTML](196) +[PDF](632.1KB) PDF Downloads(94)
Letter to the editors in chief
Tai-Ping Liu and Shih-Hsien Yu
2018, 11(1) : 215-217 doi: 10.3934/krm.2018011 +[Abstract](2663) +[HTML](292) +[PDF](273.0KB) PDF Downloads(84)
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](1140) +[HTML](288) +[PDF](477.57KB) PDF Downloads(74)
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](808) +[HTML](178) +[PDF](487.82KB) PDF Downloads(72)
Linear Boltzmann equation and fractional diffusion
Claude Bardos, François Golse and Ivan Moyano
2018, 11(4) : 1011-1036 doi: 10.3934/krm.2018039 +[Abstract](899) +[HTML](157) +[PDF](470.71KB) 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](1129) +[HTML](291) +[PDF](511.26KB) PDF Downloads(70)
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](677) +[HTML](164) +[PDF](491.35KB) PDF Downloads(68)
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](838) +[HTML](105) +[PDF](459.44KB) PDF Downloads(68)
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](766) +[HTML](200) +[PDF](1843.94KB) PDF Downloads(67)
Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary
Linjie Xiong
2018, 11(3) : 469-490 doi: 10.3934/krm.2018021 +[Abstract](957) +[HTML](213) +[PDF](497.17KB) PDF Downloads(66)

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