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Kinetic & Related Models

2009 , Volume 2 , Issue 1

A special issue on
Asymptotic Behavior of Dissipative PDEs

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Preface II
Pierangelo Marcati
2009, 2(1): v-vii doi: 10.3934/krm.2009.2.1v +[Abstract](296) +[PDF](46.0KB)
Abstract:
Angelo Marcello Anile was an internationally known applied mathematician and mathematical physicist, passed away on November 16, 2007 after one very difficult year. Angelo Marcello Anile was an internationally known applied mathematician and mathematical physicist, passed away on November 16, 2007 after one very difficult year.
I personally had the privilege to know Marcello long time ago, through our common friend and colleague Giovanni Russo, who was a former student of Marcello freshly hired that time from my Department.
I was impressed from his results on hydrodynamical models for Semiconductor Devices and, together Peter Markowich and Roberto Natalini, on December 1993 we invited him at the CNR IAC Institute Mauro Picone, where we were organizing a workshop on these topics. Marcello gave a beautiful minicourse regarding An extended thermodynamic framework for the hydrodynamical modeling of semiconductors.

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Preface I
Claude Bardos, Francois Golse and Paul Martin
2009, 2(1): i-iv doi: 10.3934/krm.2009.2.1i +[Abstract](223) +[PDF](51.9KB)
Abstract:
Basil Nicolaenko was born in Paris in 1942, to Russian immigrants. A former Cossack, his father first took a blue collar job at the nearby Renault car factory, and later became a taxi driver. Basil grew up in his parents' studio, in the southwest of Paris, where he lived until the age of 20. His high school education was supported by a scholarship of the French Government. However, things were far from simple: because of their still relatively poor command of French, his parents could hardly help him with the subtleties of the French bureaucracy. Thus, at the age of 11, Basil found himself having to argue in person with the uncooperative headmaster of Lycée Buffon to gain admission to 6th grade -- a fact he still remembered with unconcealed indignation more than 50 years later.

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Orientation waves in a director field with rotational inertia
Giuseppe Alì and John K. Hunter
2009, 2(1): 1-37 doi: 10.3934/krm.2009.2.1 +[Abstract](343) +[PDF](747.2KB)
Abstract:
We study a variational system of nonlinear hyperbolic partial differential equations that describes the propagation of orientation waves in a director field with rotational inertia and potential energy given by the Oseen-Frank energy from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves, respectively. Weakly nonlinear splay waves are described by the quadratically nonlinear Hunter-Saxton equation. In this paper, we derive a new cubically nonlinear asymptotic equation that describes weakly nonlinear twist waves. This equation provides a surprising representation of the Hunter-Saxton equation, and like the Hunter-Saxton equation it is completely integrable. There are analogous cubically nonlinear representations of the Camassa-Holm and Degasperis-Procesi equations. Moreover, two different, but compatible, variational principles for the Hunter-Saxton equation arise from a single variational principle for the primitive director field equations in the two different limits for splay and twist waves. We also use the asymptotic equation to analyze a one-dimensional initial value problem for the director-field equations with twist-wave initial data.
The multi-water-bag equations for collisionless kinetic modeling
Nicolas Besse, Florent Berthelin, Yann Brenier and Pierre Bertrand
2009, 2(1): 39-80 doi: 10.3934/krm.2009.2.39 +[Abstract](341) +[PDF](2005.8KB)
Abstract:
In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. After recalling the link of the multi-water-bag model with kinetic formulation of conservation laws, we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. These models are very promising because they reveal to be very useful for the theory and numerical simulations of laser-plasma and gyrokinetic physics. In this paper we prove some existence and uniqueness results for classical solutions of these different models. We next propose numerical schemes based on Discontinuous Garlerkin methods to solve these equations. We then present some numerical simulations of non linear problems arising in plasma physics for which we know some analytical results.
1D numerical simulation of the mep mathematical model in ballistic diode problem
Alexander Blokhin and Alesya Ibragimova
2009, 2(1): 81-107 doi: 10.3934/krm.2009.2.81 +[Abstract](340) +[PDF](832.5KB)
Abstract:
Numerical algorithms for finding approximate solutions of the macroscopic balance equations of charge transport in semiconductors based on the maximum entropy principle [A.M. Anile, V. Romano, Non parabolic band transport in semiconductors: closure of the moment equations, Contin. Mech. Thermodyn. 11 (1999), 307--325; V. Romano, Non parabolic band transport in semiconductors: closure of the production terms in the moment equations, Contin. Mech. Thermodyn. 12(2000), 31--51] are constructed and discussed for a typical 1D problem.
Stability of the travelling wave in a 2D weakly nonlinear Stefan problem
Claude-Michel Brauner, Josephus Hulshof and Luca Lorenzi
2009, 2(1): 109-134 doi: 10.3934/krm.2009.2.109 +[Abstract](298) +[PDF](328.4KB)
Abstract:
We investigate the stability of the travelling wave (TW) solution in a 2D Stefan problem, a simplified version of a solid-liquid interface model. It is intended as a paradigm problem to present our method based on: (i) definition of a suitable linear one dimensional operator, (ii) projection with respect to the $x$ coordinate only; (iii) Lyapunov-Schmidt method. The main issue is that we are able to derive a parabolic equation for the corrugated front $\varphi$ near the TW as a solvability condition. This equation involves two linear pseudo-differential operators, one acting on $\varphi$, the other on $(\varphi_y)^2$ and clearly appears as a generalization of the Kuramoto-Sivashinsky equation related to turbulence phenomena in chemistry and combustion. A large part of the paper is devoted to study the properties of these operators in the context of functional spaces in the $y$ and $x,y$ coordinates with periodic boundary conditions. Technical results are deferred to the appendices.
A Boltzmann-like equation for choice formation
Valeriano Comincioli, Lucia Della Croce and Giuseppe Toscani
2009, 2(1): 135-149 doi: 10.3934/krm.2009.2.135 +[Abstract](356) +[PDF](297.7KB)
Abstract:
We describe here a possible approach to the formation of choice in a society by methods borrowed from the kinetic theory of rarefied gases. It is shown that the evolution of the continuous density of opinions obeys a linear Boltzmann equation where the background density represents the fixed distribution of possible choices. The binary interactions between individuals are in general non-local, and take into account both the compromise propensity and the self-thinking. In particular regimes, the linear Boltzmann equation is well described by a Fokker-Planck type equation, for which in some cases the steady states (distribution of choices) can be obtained in analytical form. This Fokker-Planck type equation generalizes analogous one obtained by mean field approximation of the voter model in [27]. Numerical examples illustrate the influence of different model parameters in the description both of the shape of the distribution of choices, and in its mean value.
On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation
Bernard Ducomet, Alexander Zlotnik and Ilya Zlotnik
2009, 2(1): 151-179 doi: 10.3934/krm.2009.2.151 +[Abstract](491) +[PDF](582.4KB)
Abstract:
We consider a 1D Schrödinger equation with variable coefficients on the half-axis. We study a family of two-level symmetric finite-difference schemes with a three-point parameter dependent averaging in space. This family includes a number of particular schemes. The schemes are coupled to an approximate transparent boundary condition (TBC). We prove two stability bounds with respect to initial data and a free term in the main equation, under suitable conditions on an operator of the approximate TBC. We also consider the family of schemes on an infinite mesh in space. We derive and analyze the discrete TBC allowing to restrict these schemes to a finite mesh and prove the stability conditions for it. Numerical examples are also included.
A note on the time decay of solutions for the linearized Wigner-Poisson system
Irene M. Gamba, Maria Pia Gualdani and Christof Sparber
2009, 2(1): 181-189 doi: 10.3934/krm.2009.2.181 +[Abstract](378) +[PDF](149.0KB)
Abstract:
We consider the one-dimensional Wigner-Poisson system of plasma physics, linearized around a (spatially homogeneous) Lorentzian distribution and prove that the solution of the corresponding linearized problem decays to zero in time. We also give an explicit algebraic decay rate.
Modelling and simulation of a solar updraft tower
Ingenuin Gasser
2009, 2(1): 191-204 doi: 10.3934/krm.2009.2.191 +[Abstract](402) +[PDF](822.0KB)
Abstract:
A new model for a solar updraft tower is presented. It is based on a one-dimensional description of the fully transient gasdynamics in an updraft power plant from the outer end of the collector to the top of the tower. All the main physical effects are included. The model is derived from basic gasdynamic equations, a low Mach number asymptotics is performed and numerical simulations are shown.
Local Hilbert expansion for the Boltzmann equation
Yan Guo, Juhi Jang and Ning Jiang
2009, 2(1): 205-214 doi: 10.3934/krm.2009.2.205 +[Abstract](307) +[PDF](158.4KB)
Abstract:
We revisit the classical work of Caflisch [1] for compressible Euler limit of the Boltzmann equation. By using a new $L^{2}\mbox{-}L^{\infty }$ method, we prove the validity of the Hilbert expansion before shock formations in the Euler system with moderate temperature variation.
Three-dimensional instabilities in non-parallel shear stratified flows
Alex Mahalov, Mohamed Moustaoui and Basil Nicolaenko
2009, 2(1): 215-229 doi: 10.3934/krm.2009.2.215 +[Abstract](274) +[PDF](2752.6KB)
Abstract:
The instabilities of non-parallel flows ($\overline{U}(x_3)$, $\overline{V}(x_3), 0)$ ($\overline{V} \ne 0$) such as those induced by polarized inertia-gravity waves embedded in a stably stratified environment are analyzed in the context of the 3D Euler-Boussinesq equations. We derive a sufficient condition for shear stability and a necessary condition for instability in the case of non-parallel velocity fields. Three dimensional numerical simulations of the full nonlinear equations are conducted to characterize the respective modes of instability, their topology and dynamics, and subsequent breakdown into turbulence. We describe fully three-dimensional instability mechanisms, and study spectral properties of the most unstable modes. Our stability/instability criteria generalizes that in the case of parallel shear flows ($\bar{V}=0$), where stability properties are governed by the Taylor-Goldstein equations previously studied in the literature. Unlike the case of parallel flows, the polarized horizontal velocity vector rotating with respect to the vertical coordinate ($x_3$) excites unstable modes that have different spectral properties depending on the orientation of the velocity vector. At each vertical level, the horizontal wave vector of the fastest growing mode is parallel to the local vector ($ d\overline{U}(x_3)/dx_3$, $d \overline{V}(x_3)/dx_3)$. We investigate three-dimensional characteristics of the unstable modes and present computational results on Lagrangian particle dynamics.
Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics
Giovanni Russo and Francis Filbet
2009, 2(1): 231-250 doi: 10.3934/krm.2009.2.231 +[Abstract](298) +[PDF](923.5KB)
Abstract:
In this paper we present a new semilagrangian scheme for the numerical solution of the BGK model of rarefied gas dynamics, in a domain with moving boundaries, in view of applications to Micro Electro Mechanical Systems (MEMS). The source term is treated implicitly, which makes the scheme Asymptotic Preserving in the limit of small Knudsen number. Because of its Lagrangian nature, no stability restriction is posed on the CFL number, which is determined only by accuracy requirements. The method is tested on a one dimensional piston problem. The solution for small Knudsen number is compared with the results obtained by the numerical solution of the Euler equation of gas dynamics.

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