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Kinetic & Related Models

2013 , Volume 6 , Issue 4

Issue dedicated to Petr P. Zabreiko on the occasion of his 70th birthday

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Seiji Ukai
Yan Guo
2013, 6(4): iii-iii doi: 10.3934/krm.2013.6.4iii +[Abstract](190) +[PDF](80.8KB)
Abstract:
Seiji Ukai has made fundamental contributions to the mathematical study for the Boltzmann equation. His seminal construction of global solutions near Maxwellian opens the era for the study of non-homogeneous Boltzmann solutions, his work on a gas passing an obstacle remains a penetrating classical result, and his study of inverse power laws without angular cutoff marks a breakthrough in the field.
    I first met Seiji in 1995 at Oberwolfach. I remember that he smoked a lot and we had to stand in the cold for discussions. He was gentle and soft-spoken, paused and thought carefully before answering my questions. We met several years later again at Oberwolfach, when I presented my construction of global smooth solutions to the compressible Euler-Poisson system. He was excited and relaxed, and with Bob Glassey, we talked a lot about problems in fluids.
    Motivated by my desire to understand collision effects in a plasma, I decided to learn and work on the Boltzmann equation. I was greatly influenced by his elegant survey paper `Solutions of the Boltzmann Equation', and later found a different way to control the macroscopic part of Boltzmann solutions. I was very encouraged by the positive feedbacks from the Boltzmann community, in particular, Desvillettes, Illner, Ukai and Villani, among others.
    In 2001, I visited Japan, passing through Tokyo. Seiji insisted meeting with me at my hotel in Tokyo (he had to travel for two hours for that!). He took me out for dinner and we chatted a lot about everything. As with many of my Japanese friends, when conversations go deeper, sometimes we resort to Chinese characters for better communications. His knowledge in Chinese characters was so impressive, and I remember clearly our discussion about the Chinese character `head'. He was shocked to learn its simplified form currently in use, which is so much different from its traditional counterpart. Shooking his head in disbelief, he wanted to know the exact reason behind such a simplification. A true scholar, Seiji was so meticulous about every detail! It got quite late and I stood outside my hotel, watching him walking quickly towards the subway station.
    I have not seen Seiji too much in recent years. I am always grateful for his encouragements and mathematical insights, and will always remember the lovely evening we spent together in 2001.
Preface
Kazuo Aoki, Pierre Degond and Tong Yang
2013, 6(4): i-ii doi: 10.3934/krm.2013.6.4i +[Abstract](192) +[PDF](504.2KB)
Abstract:
In 2012, we are saddened by the loss of our friend and a great mathematician, Seiji Ukai, who passed away in November. Seiji was a leading expert in the area of kinetic theory and fluid mechanics, and was always kind and humble so that he was beloved by his colleagues, collaborators and researchers in these research areas. As Cédric Villani said in his article, ``Good-bye my friend...(In Memoriam Seiji Ukai)", ``He was a great mathematician and a beautiful soul".

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A Milne problem from a Bose condensate with excitations
Leif Arkeryd and Anne Nouri
2013, 6(4): 671-686 doi: 10.3934/krm.2013.6.671 +[Abstract](297) +[PDF](401.1KB)
Abstract:
This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead we use the energy flow which is constant, but no more hydrodynamic.
Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force
Renjun Duan and Shuangqian Liu
2013, 6(4): 687-700 doi: 10.3934/krm.2013.6.687 +[Abstract](345) +[PDF](383.7KB)
Abstract:
We are concerned with a two-phase flow system consisting of the Vlasov-Fokker-Planck equation for particles coupled to the compressible Euler equations for the fluid through the friction force. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The proof is based on the classical energy estimates.
Unstable galaxy models
Zhiyu Wang, Yan Guo, Zhiwu Lin and Pingwen Zhang
2013, 6(4): 701-714 doi: 10.3934/krm.2013.6.701 +[Abstract](328) +[PDF](489.3KB)
Abstract:
The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of $\Phi(E,L)$, where $E$ is the particle energy and $L$ the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem [7], the galaxy model $\Phi(E,L)$ is stable if the distribution $\Phi$ is monotonically decreasing with respect to the particle energy $E.$ On the other hand, the stability of $\Phi(E,L)$ remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin [11], we constuct examples of unstable galaxy models of $f(E,L)$ and $f\left( E\right) \ $in which $f$ fails to be monotone in $E.$
A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu
2013, 6(4): 715-727 doi: 10.3934/krm.2013.6.715 +[Abstract](232) +[PDF](386.5KB)
Abstract:
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class $S_{1/2}^{1/2}(\mathbb{R}^d)$, implying the ultra-analyticity and the production of exponential moments of the fluctuation, for any positive time.
Global stability of stationary waves for damped wave equations
Lili Fan, Hongxia Liu, Huijiang Zhao and Qingyang Zou
2013, 6(4): 729-760 doi: 10.3934/krm.2013.6.729 +[Abstract](312) +[PDF](502.2KB)
Abstract:
Nonlinear stability of stationary waves to damped wave equations has been studied by many authors in recent years, the main difficulty lies in how to control the possible growth of its solutions caused by the nonlinearity of the equation under consideration. An effective way to overcome such a difficulty is to use the smallness of the initial perturbation and/or the smallness of the strength of the stationary waves and based on this argument, local stability of strong increasing stationary waves for convex flux functions and weak decreasing stationary waves for general flux functions are well-established, cf. [11,13,15]. As to the nonlinear stability result with large initial perturbation, the only results available now are [3,4] which use the smallness of the stationary waves to overcome the above mentioned difficulty and consequently, although the initial perturbation can be large, it does require that it satisfies certain growth condition as the strength of the stationary waves tends to zero. Thus a natural question of interest is, for any initial perturbation lying in certain Sobolev space $H^s\left({\bf R}_+\times{\bf R}^{n-1}\right)$, how to obtain the global stability of stationary waves to the damped wave equations? The main purpose of this manuscript is devoted to this problem.
Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval
Raffaele Esposito, Yan Guo and Rossana Marra
2013, 6(4): 761-787 doi: 10.3934/krm.2013.6.761 +[Abstract](310) +[PDF](523.0KB)
Abstract:
We consider a kinetic model for a system of two species of particles on a sufficiently large periodic interval, interacting through a long range repulsive potential and by collisions. The model is described by a set of two coupled Vlasov-Boltzmann equations. For temperatures below the critical value and suitably prescribed masses, there is a non homogeneous solution, the double soliton, which is a minimizer of the entropy functional. We prove the stability, up to translations, of the double soliton under small perturbations. The same arguments imply the stability of the pure phases, as well as the stability of the mixed phase above the critical temperature. The mixed phase is proved to be unstable below the critical temperature.
Transport coefficients in the $2$-dimensional Boltzmann equation
Alexander Bobylev and Raffaele Esposito
2013, 6(4): 789-800 doi: 10.3934/krm.2013.6.789 +[Abstract](289) +[PDF](345.7KB)
Abstract:
We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the $2$-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.
On self-similar solutions to the homogeneous Boltzmann equation
Marco Cannone and Grzegorz Karch
2013, 6(4): 801-808 doi: 10.3934/krm.2013.6.801 +[Abstract](332) +[PDF](333.5KB)
Abstract:
In this note, our results from [ Comm. Pure Appl. Math. 63 (2010), 747--778] on infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules are discussed, presented in a different context, and improved by using recent observations by Morimoto and Yang. In particular, similarities between the homogeneous Boltzmann equation and the fractional heat equation are emphasized. Moreover, we show that a certain conjecture by Bobylev and Cercignani on regularity of self-similar solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules has a positive answer.
Vision-based macroscopic pedestrian models
Pierre Degond, Cécile Appert-Rolland, Julien Pettré and Guy Theraulaz
2013, 6(4): 809-839 doi: 10.3934/krm.2013.6.809 +[Abstract](368) +[PDF](746.9KB)
Abstract:
We propose a hierarchy of kinetic and macroscopic models for a system consisting of a large number of interacting pedestrians. The basic interaction rules are derived from [44] where the dangerousness level of an interaction with another pedestrian is measured in terms of the derivative of the bearing angle (angle between the walking direction and the line connecting the two subjects) and of the time-to-interaction (time before reaching the closest distance between the two subjects). A mean-field kinetic model is derived. Then, three different macroscopic continuum models are proposed. The first two ones rely on two different closure assumptions of the kinetic model, respectively based on a monokinetic and a von Mises-Fisher distribution. The third one is derived through a hydrodynamic limit. In each case, we discuss the relevance of the model for practical simulations of pedestrian crowds.
On a voltage-conductance kinetic system for integrate & fire neural networks
Benoît Perthame and Delphine Salort
2013, 6(4): 841-864 doi: 10.3934/krm.2013.6.841 +[Abstract](311) +[PDF](481.0KB)
Abstract:
The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network. It is used when slow conductance receptors are activated and noticeable applications to the visual cortex have been worked-out. In the simplest case, the derivation also uses the assumption of fully excitatory and moderately all-to-all coupled networks; this is the situation we consider here.
    We study properties of solutions of the kinetic equation for steady states and time evolution and we prove several global a priori bounds both on the probability density and the firing rate of the network. The main difficulties are related to the degeneracy of the diffusion resulting from noise and to the quadratic aspect of the nonlinearity.
    This result constitutes a paradox; the solutions of the kinetic model, of partially hyperbolic nature, are globally bounded but it has been proved that the fully parabolic integrate and fire equation (some kind of diffusion limit of the former) blows-up in finite time.
Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition
Shijin Deng, Linglong Du and Shih-Hsien Yu
2013, 6(4): 865-882 doi: 10.3934/krm.2013.6.865 +[Abstract](268) +[PDF](409.8KB)
Abstract:
This is a continuation of the paper [5] on the Broadwell model with conservative boundary condition. In this paper, based on the full boundary data obtained by LY algorithm and recombination lemma, we construct the Green's function for the initial boundary value problem. We also establish the pointwise convergence estimate of the solution and nonlinear stability of the global Maxwellian under sufficiently small initial perturbation.
Energy estimate for a linear symmetric hyperbolic-parabolic system in half line
Tohru Nakamura and Shinya Nishibata
2013, 6(4): 883-892 doi: 10.3934/krm.2013.6.883 +[Abstract](311) +[PDF](340.2KB)
Abstract:
In the present paper, we study the initial boundary value problem for a linear symmetric hyperbolic-parabolic system in one-dimensional half space. We obtain a priori estimates by using an energy method developed by Matsumura--Nishida for half space problem under the assumption that a stability condition of Shizuta--Kawashima type holds. The method developed in the present paper is applicable to showing the nonlinear stability of boundary layer solutions for a system of viscous conservation laws in half space.
The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits
Claude Bardos and Nicolas Besse
2013, 6(4): 893-917 doi: 10.3934/krm.2013.6.893 +[Abstract](261) +[PDF](494.7KB)
Abstract:
This contribution concerns a one-dimensional version of the Vlasov equation dubbed the Vlasov$-$Dirac$-$Benney equation (in short V$-$D$-$B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with the so-called Benney equation leads to new stability results. Eventually the V$-$D$-$B appears to be at the ``cross road" of several problems of mathematical physics which have as far as stability is concerned very similar properties.
Empirical measures and Vlasov hierarchies
François Golse, Clément Mouhot and Valeria Ricci
2013, 6(4): 919-943 doi: 10.3934/krm.2013.6.919 +[Abstract](254) +[PDF](526.5KB)
Abstract:
The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the $N$-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent $1$ on the infinite mean field hierarchy. This last result amplifies Spohn's uniqueness theorem [H. Spohn, H. Neunzert, Math. Meth. Appl. Sci. 3 (1981), 445--455].
Convex analysis and thermodynamics
Nelly Point and Silvano Erlicher
2013, 6(4): 945-954 doi: 10.3934/krm.2013.6.945 +[Abstract](275) +[PDF](312.2KB)
Abstract:
Convex analysis is very useful to prove that a material model fulfills the second law of thermodynamics. Dissipation must remains non-negative and an elegant way to ensure this property is to construct an appropriate pseudo-potential of dissipation. In such a case, the corresponding material is said to be a Standard Generalized Material and the flow rules fulfill a normality rule (i.e. the dissipative thermodynamic forces are assumed to belong to an admissible domain and the flow of the corresponding state variables is orthogonal to the boundary of this domain). The sum of the pseudo-potential with its Legendre-Fenchel conjugate fulfills the Fenchel's inequality and as the actual value of the dual pair forces-flows minimizes this inequality, this can be used as a convergence criterium for numerical applications. Actually, some very commonly used and effective models do not fit into that family of Standard Generalized Materials. A procedure is here proposed which permits to retrieve the normality assumption and to construct a pair of dual pseudo-potentials also for these non-standard material models. This procedure was first presented by the authors for non-associated plasticity. Now it is extended to a large range of mechanical problems.
Some properties of the kinetic equation for electron transport in semiconductors
Wolfgang Wagner
2013, 6(4): 955-967 doi: 10.3934/krm.2013.6.955 +[Abstract](270) +[PDF](372.3KB)
Abstract:
The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.
Asymptotic behavior of solutions to the generalized cubic double dispersion equation in one space dimension
Masakazu Kato, Yu-Zhu Wang and Shuichi Kawashima
2013, 6(4): 969-987 doi: 10.3934/krm.2013.6.969 +[Abstract](304) +[PDF](470.0KB)
Abstract:
We study the initial value problem for the generalized cubic double dispersion equation in one space dimension. We establish a nonlinear approximation result to our global solutions that was obtained in [6]. Moreover, we show that as time tends to infinity, the solution approaches the superposition of nonlinear diffusion waves which are given explicitly in terms of the self-similar solution of the viscous Burgers equation. The proof is based on the semigroup argument combined with the analysis of wave decomposition.
Remarks on the full dispersion Kadomtsev-Petviashvli equation
David Lannes and Jean-Claude Saut
2013, 6(4): 989-1009 doi: 10.3934/krm.2013.6.989 +[Abstract](287) +[PDF](4149.4KB)
Abstract:
We consider in this paper the Full Dispersion Kadomtsev-Petviashvili Equation (FDKP) introduced in [19] in order to overcome some shortcomings of the classical KP equation. We investigate its mathematical properties, emphasizing the differences with the Kadomtsev-Petviashvili equation and their relevance to the approximation of water waves. We also present some numerical simulations.
Local existence with mild regularity for the Boltzmann equation
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu and Tong Yang
2013, 6(4): 1011-1041 doi: 10.3934/krm.2013.6.1011 +[Abstract](309) +[PDF](559.6KB)
Abstract:
Without Grad's angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the angular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators.
A kinetic description of mutation processes in bacteria
Giuseppe Toscani
2013, 6(4): 1043-1055 doi: 10.3934/krm.2013.6.1043 +[Abstract](277) +[PDF](355.4KB)
Abstract:
The Luria--Delbrück mutation model has been mathematically formulated in a number of ways. Last, a mean field picture derived from a kinetic formulation has been derived by Kashdan and Pareschi in [18]. There, the Luria--Delbrück distribution appears as the solution of a Fokker-Planck like equation obtained as the quasi-invariant asymptotics of a linear Boltzmann equation for the number density of the number of mutated cells. This paper addresses the kinetic description for the Lea--Coulson formulation [21], as well as for the Kendall formulation [19], focusing on important modeling issues closely linked with the distribution of the number of mutants. The paper additionally emphasizes basic principles which not only help to unify existing results but also allow for a useful extensions.

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