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Kinetic & Related Models

2014 , Volume 7 , Issue 1

Issue on analysis of non-equilibrium evolution problems: Selected topics in material and life sciences

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A note from the Editorial Board
Editorial Board
2014, 7(1): i-i doi: 10.3934/krm.2014.7.1i +[Abstract](65) +[PDF](77.3KB)
Robert Glassey (Bob) has decided to withdraw from the Editorial Board of KRM. Bob has been a pioneer of mathematical kinetic theory in the 80's and one of its leading figures since then. His seminal papers on the relativistic Vlasov-Poisson and Vlasov-Maxwell systems and their asymptotic stability have been a source of inspiration for many of us. His book, `The Cauchy problem in kinetic theory' has become a must for all young researchers entering the field. Bob has been involved in the Editorial Board of KRM since the beginning of the journal and has contributed to the edition of many papers. On behalf of the whole editorial board, we express our deep gratitude to him for having joined us in this adventure and contributed to the success of the journal.
A kinetic equation for spin polarized Fermi systems
Leif Arkeryd
2014, 7(1): 1-8 doi: 10.3934/krm.2014.7.1 +[Abstract](130) +[PDF](336.1KB)
This paper considers a kinetic Boltzmann equation, having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures. The distribution functions have values in the space of positive hermitean $2\times2$ complex matrices. Global existence of weak solutions is proved in $L^1\cap L^{\infty}$ for the initial value problem of this Boltzmann equation in a periodic box.
Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$
Shen Bian, Jian-Guo Liu and Chen Zou
2014, 7(1): 9-28 doi: 10.3934/krm.2014.7.9 +[Abstract](109) +[PDF](481.3KB)
This paper establishes the hyper-contractivity in $L^\infty(\mathbb{R}^d)$ (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent $m>1-2/d$. The results show that for the supercritical and critical case $1-2/d < m ≤ 2-2/d$, if $||U_0||_{d(2-m)/2} < C_{d,m}$ where $C_{d,m}$ is a universal constant, then for any $t>0$, $||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ is bounded and decays as $t$ goes to infinity. For the subcritical case $m>2-2/d$, the solution $u(\cdot,t) \in L^\infty(\mathbb{R}^d)$ with any initial data $U_0 \in L_+^1(\mathbb{R}^d)$ for any positive time.
A mathematical model for value estimation with public information and herding
Marcello Delitala and Tommaso Lorenzi
2014, 7(1): 29-44 doi: 10.3934/krm.2014.7.29 +[Abstract](150) +[PDF](1765.7KB)
This paper deals with a class of integro-differential 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 well-posedness 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 over-exponential rise of the value estimated by the market, paving the way to the formation of economic bubbles.
Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion
Jishan Fan and Tohru Ozawa
2014, 7(1): 45-56 doi: 10.3934/krm.2014.7.45 +[Abstract](102) +[PDF](361.5KB)
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.
Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation
Yueling Jia and Zhaohui Huo
2014, 7(1): 57-77 doi: 10.3934/krm.2014.7.57 +[Abstract](124) +[PDF](508.6KB)
The inviscid limit behavior of solution is considered for the multi-dimensional derivative complex Ginzburg-Landau(DCGL) equation. For small initial data, it is proved that for some $T>0$, solution of the DCGL equation converges to the solution of the derivative nonlinear Schrödinger (DNLS) equation in natural space $C([0,T]; H^s)(s\geq \frac{n}{2})$ if some coefficients tend to zero.
Long time asymptotics of a degenerate linear kinetic transport equation
Tomasz Komorowski
2014, 7(1): 79-108 doi: 10.3934/krm.2014.7.79 +[Abstract](84) +[PDF](577.6KB)
In the present article we prove an algebraic rate of decay towards the equilibrium for the solution of a non-homogeneous, linear kinetic transport equation. The estimate is of the form $C(1+t)^{-a}$ for some $a>0$. The total scattering cross-section $R(k)$ is allowed to degenerate but we assume that $R^{-a}(k)$ is integrable with respect to the invariant measure.
Stability of solutions of kinetic equations corresponding to the replicator dynamics
Mirosław Lachowicz, Andrea Quartarone and Tatiana V. Ryabukha
2014, 7(1): 109-119 doi: 10.3934/krm.2014.7.109 +[Abstract](105) +[PDF](506.5KB)
In the present paper we propose a class of kinetic type equations that describes the replicator dynamics at the mesoscopic level. The equations are highly nonlinear due to the dependence of the transition rates of distribution function. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations.
Decay of solutions to generalized plate type equations with memory
Shikuan Mao and Yongqin Liu
2014, 7(1): 121-131 doi: 10.3934/krm.2014.7.121 +[Abstract](114) +[PDF](385.1KB)
In this paper we focus on the initial value problem of an inertial model for a generalized plate equation with memory in $\mathbb{R}^n\ (n\geq1)$. We study the decay and the regularity-loss property for this type of equations in the spirit of [10,13]. The novelty of this paper is that we extend the order of derivatives from integer to fraction and refine the results of the even-dimensional case in the related literature [10,13].
Thermodynamical considerations implying wall/particles scattering kernels
Vincent Pavan
2014, 7(1): 133-168 doi: 10.3934/krm.2014.7.133 +[Abstract](77) +[PDF](573.0KB)
In this paper we study the thermodynamics of a rarefied gas contained in a closed vessel at constant volume. By adding axiomatic rules to the usual ones derived by Cercignani, we obtain a new symmetry property in the wall/particle scattering kernel. This new symmetry property enables us to show the first and second law of macroscopic thermodynamics for a rarefied gas having collisions with walls. Then we study the behavior of the rarefied gas when it is in contact with several (moving) thermostats at the same time. We show the existence, uniqueness and long time behavior of the solution to the homogeneous (linear) evolution equation describing the system. Finally we apply our thermodynamical model of rarefied gas to the measurement of heat flux in very low density systems and compare it to the experimental results, shading into light new interpretation of observed behaviors.
Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation
Linjie Xiong, Tao Wang and Lusheng Wang
2014, 7(1): 169-194 doi: 10.3934/krm.2014.7.169 +[Abstract](178) +[PDF](526.3KB)
The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.
Blowup of smooth solutions to the full compressible MHD system with compact density
Baoquan Yuan and Xiaokui Zhao
2014, 7(1): 195-203 doi: 10.3934/krm.2014.7.195 +[Abstract](110) +[PDF](347.2KB)
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.

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