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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.
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 KRM is a publication of the American Institute of Mathematical Sciences. All rights reserved.

TOP 10 Most Read Articles in KRM, July 2017
1 
Kinetic theory and numerical simulations of twospecies coagulation
Volume 7, Number 2, Pages: 253  290, 2014
Carlos Escudero,
Fabricio Macià,
Raúl Toral
and Juan J. L. Velázquez
Abstract
References
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In this work we study the stochastic process of twospecies
coagulation. This process consists in the aggregation dynamics
taking place in a ring. Particles and clusters of particles are set
in this ring and they can move either clockwise or counterclockwise.
They have a probability to aggregate forming larger clusters when
they collide with another particle or cluster. We study the
stochastic process both analytically and numerically. Analytically,
we derive a kinetic theory which approximately describes the process
dynamics. One of our strongest assumptions in this respect
is the so called wellstirred limit, that allows neglecting the
appearance of spatial coordinates in the theory, so this becomes
effectively reduced to a zeroth dimensional model.
We determine the long time behavior of such a model, making emphasis
in one special case in which it displays selfsimilar solutions.
In particular these calculations
answer the question of how the system gets ordered, with all
particles and clusters moving in the same direction, in the long
time. We compare our analytical results with direct numerical
simulations of the stochastic process and both corroborate its
predictions and check its limitations. In particular, we numerically
confirm the ordering dynamics predicted by the kinetic theory and
explore properties of the realizations of the stochastic process
which are not accessible to our theoretical approach.

2 
Regularity criteria for the 3D MHD equations via partial derivatives. II
Volume 7, Number 2, Pages: 291  304, 2014
Xuanji Jia
and Yong Zhou
Abstract
References
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In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1,3,9,11,17]. Precisely, we first prove that if for any $ i,\,j,\,k\in \{1,2,3\}$ there holds
$
(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k}) \in L_T^{\alpha,\gamma}$ with $\frac{2}{\alpha}+\frac{3}{\gamma}\leq 1+\frac{1}{\gamma},~2\leq \gamma\leq \infty
$, then the solution $(u,b)$ is smooth on $\mathbb{R}^3\times(0,T]$. Secondly, we show that any component (resp. components) of $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k})$ in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space $L_T^{\alpha',\gamma'}$with $\frac{2}{\alpha'}+\frac{3}{\gamma'}\leq 1$, $3< \gamma'\leq \infty$. Fianlly, we obtain a LadyzhenskayaProdiSerrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].

3 
Gassurface interaction and boundary conditions for the Boltzmann equation
Volume 7, Number 2, Pages: 219  251, 2014
Stéphane Brull,
Pierre Charrier
and Luc Mieussens
Abstract
References
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In this paper we revisit the derivation of boundary
conditions for the Boltzmann Equation. The interaction between the
wall atoms and the gas molecules within a thin surface layer is
described by a kinetic equation introduced in [10] and used
in [1]. This equation includes a Vlasov term and a linear
moleculephonon collision term and is coupled with the Boltzmann
equation describing the evolution of the gas in the bulk
flow. Boundary conditions are formally derived from this model by
using classical tools of kinetic theory such as scaling and systematic
asymptotic expansion. In a first step this method is applied to the
simplified case of a flat wall. Then it is extented to walls with
nanoscale roughness allowing to obtain more complex scattering
patterns related to the morphology of the wall. It is proved that the
obtained scattering kernels satisfy the classical imposed properties
of nonnegativeness, normalization and reciprocity introduced by
Cercignani [13].

4 
A mathematical model for value estimation with public information and herding
Volume 7, Number 1, Pages: 29  44, 2013
Marcello Delitala
and Tommaso Lorenzi
Abstract
References
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This paper deals with a class of integrodifferential equations modeling the dynamics of a market where agents estimate the value of a given traded good. Two basic mechanisms are assumed to concur in value estimation: interactions between agents and sources of public information and herding phenomena. A general wellposedness result is established for the initial value problem linked to the model and the asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic different economic scenarios. Analytical results are illustrated by means of numerical simulations and lead us to conclude that, in spite of its oversimplified nature, this model is able to reproduce some emerging behaviors proper of the system under consideration. In particular, consistently with experimental evidence, the obtained results suggest that if agents are highly confident in the product, imitative and scarcely rational behaviors may lead to an overexponential rise of the value estimated by the market, paving the way to the formation of economic bubbles.

5 
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.

6 
A random cloud model for the Schrödinger equation
Volume 7, Number 2, Pages: 361  379, 2014
Wolfgang Wagner
Abstract
References
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The paper is concerned with the construction of a
stochastic model for the spatially discretized
timedependent Schrödinger equation.
The model is based on a particle system with
a Markov jump evolution.
The particles are characterized by a sign (plus or minus),
a position (discrete grid) and a type (real or imaginary).
The jumps are determined by the creation of offspring.
The main result is the construction of a family of complexvalued random
variables such that their expected
values coincide with the solution of the
Schrödinger equation.

7 
Stability and modeling error for the Boltzmann equation
Volume 7, Number 2, Pages: 401  414, 2014
El Miloud Zaoui
and Marc Laforest
Abstract
References
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We show that the residual measures the difference in $L^1$ between the solutions to two different Boltzmann models
of rarefied gases. This work is an extension of earlier work by Ha on the stability of Boltzmann's model, and more specifically
on a nonlinear interaction functional that controls the growth of waves. The two kinetic models that are
compared in this research are given by (possibly different) inverse power laws, such as the
hard spheres and pseudoMaxwell models.
The main point of the estimate is that the modeling error is measured a posteriori, that is to say, the difference between
the solutions to the first and second model can be bounded by a term that depends on only one of the two solutions.
This work allows the stability estimate to be used to assess
uncertainty, modeling or numerical, present in the solution of the first model without
solving the second model.

8 
Blowup of smooth solutions to the full compressible MHD system with compact density
Volume 7, Number 1, Pages: 195  203, 2013
Baoquan Yuan
and Xiaokui Zhao
Abstract
References
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This paper studies the blowup of smooth solutions to the full compressible MHD system with zero resistivity on $\mathbb{R}^{d}$, $d\geq 1$. We obtain that the smooth solutions to the MHD system will blow up in finite time, if the initial density is compactly supported.

9 
Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal
magnetic diffusion
Volume 7, Number 1, Pages: 45  56, 2013
Jishan Fan
and Tohru Ozawa
Abstract
References
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This paper proves some regularity criteria for the 2D MHD system with horizontal dissipation and horizontal
magnetic diffusion. We also prove the global existence of strong solutions of its regularized MHD$\alpha$ system.

10 
On a threeComponent CamassaHolm equation with peakons
Volume 7, Number 2, Pages: 305  339, 2014
Yongsheng Mi
and Chunlai Mu
Abstract
References
Full Text
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In this paper, we are concerned with threeComponent CamassaHolm equation with peakons. First, We establish the local wellposedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using LittlewoodPaley decomposition and transport equation theory. Second, the local wellposedness in critical case
(with $s=\frac{3}{2}, p=2,r=1$) is considered.
Then, with analytic initial data, we show
that its solutions are analytic in both variables, globally in space and
locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension
results for functions on the halfline and several symmetry preserving properties of
the equations under discussion.

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