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Kinetic & Related Models

2011 , Volume 4 , Issue 4

Issue on geometric properties for parabolic and elliptic PDE's

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Preface
Paola Pietra, Eric Polizzi, Fabrice Deluzet, Jihène Kéfi, Olivier Pinaud, Claudia Negulescu, Nicolas Vauchelet, Raymond El Hajj, Clément Jourdana and Stefan Possanner
2011, 4(4): i-iii doi: 10.3934/krm.2011.4.4i +[Abstract](195) +[PDF](892.5KB)
Abstract:
On July 5-th 2010, Naoufel Ben Abdallah tragically passed away at the age of 41. He was an extremely talented mathematician with a deep interest in applications, and an uncommon ability in creating links for interdisciplinary research.

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Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential
Naoufel Ben Abdallah, Yongyong Cai, Francois Castella and Florian Méhats
2011, 4(4): 831-856 doi: 10.3934/krm.2011.4.831 +[Abstract](331) +[PDF](1128.3KB)
Abstract:
We consider the three dimensional Gross-Pitaevskii equation\break (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confined in vertical $z$ direction. The confining potential induces high oscillations in time. If the confinement in the $z$ direction is a harmonic trap -- an approximation which is widely used in physical experiments -- the very special structure of the spectrum of the confinement operator implies that the oscillations are periodic in time. Based on this observation, it can be proved that the GPE can be averaged out with an error of order of $\epsilon$, which is the typical period of the oscillations. In this article, we construct a more accurate averaged model, which approximates the GPE up to errors of order $\mathcal{O}(\epsilon^2)$. Then, expansions of this model over the eigenfunctions (modes) of the confining operator $H_z$ in the $z$-direction are given in view of numerical applications. Efficient numerical methods are constructed to solve the GPE with cylindrical symmetry in 3D and the approximation model with radial symmetry in 2D, and numerical results are presented for various kinds of initial data.
On the minimization problem of sub-linear convex functionals
Naoufel Ben Abdallah, Irene M. Gamba and Giuseppe Toscani
2011, 4(4): 857-871 doi: 10.3934/krm.2011.4.857 +[Abstract](301) +[PDF](366.6KB)
Abstract:
The study of the convergence to equilibrium of solutions to Fokker-Planck type equations with linear diffusion and super-linear drift leads in a natural way to a minimization problem for an energy functional (entropy) which relies on a sub-linear convex function. In many cases, conditions linked both to the non-linearity of the drift and to the space dimension allow the equilibrium to have a singular part. We present here a simple proof of existence and uniqueness of the minimizer in the two physically interesting cases in which there is the constraint of mass, and the constraints of both mass and energy. The proof includes the localization in space of the (eventual) singular part. The major example is related to the Fokker-Planck equation introduced in [6, 7] to describe the evolution of both Bose-Einstein and Fermi-Dirac particles.
Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach
Naoufel Ben Abdallah, Antoine Mellet and Marjolaine Puel
2011, 4(4): 873-900 doi: 10.3934/krm.2011.4.873 +[Abstract](331) +[PDF](499.7KB)
Abstract:
We develop a Hilbert expansion approach for the derivation of fractional diffusion equations from the linear Boltzmann equation with heavy tail equilibria.
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](606) +[PDF](409.3KB)
Abstract:
This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.
Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu and Tong Yang
2011, 4(4): 919-934 doi: 10.3934/krm.2011.4.919 +[Abstract](319) +[PDF](421.8KB)
Abstract:
In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.
Stagnation-point flow of a rarefied gas impinging obliquely on a plane wall
Kazuo Aoki and Yoshiaki Abe
2011, 4(4): 935-954 doi: 10.3934/krm.2011.4.935 +[Abstract](332) +[PDF](686.2KB)
Abstract:
The steady two-dimensional stagnation-point flow of a rarefied gas impinging obliquely on an infinitely wide plane wall is investigated on the basis of kinetic theory. Assuming that the overall flow field has a length scale of variation much longer than the mean free path of the gas molecules and that the Mach number based on the characteristic flow speed is as small as the Knudsen number (the mean free path divided by the overall length scale of variation), one can exploit the result of the asymptotic theory (weakly nonlinear theory) for the Boltzmann equation, developed by Sone, that describes general steady behavior of a slightly rarefied gas over a smooth boundary [Y. Sone, in: D. Dini (ed.) Rarefied Gas Dynamics, Vol. 2, pp. 737--749. Editrice Tecnico Scientifica, Pisa (1971)]. By solving the fluid-dynamic system of equations given by the theory, the precise description of the velocity and temperature fields around the plane wall is obtained.
Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system
Blanca Ayuso, José A. Carrillo and Chi-Wang Shu
2011, 4(4): 955-989 doi: 10.3934/krm.2011.4.955 +[Abstract](412) +[PDF](624.5KB)
Abstract:
We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.
Asymptotic-preserving scheme for a bi-fluid Euler-Lorentz model
Stéphane Brull, Pierre Degond, Fabrice Deluzet and Alexandre Mouton
2011, 4(4): 991-1023 doi: 10.3934/krm.2011.4.991 +[Abstract](337) +[PDF](1125.3KB)
Abstract:
The present work is devoted to the simulation of a strongly magnetized plasma considered as a mixture of an ion fluid and an electron fluid. For the sake of simplicity, we assume that the model is isothermal and described by Euler equations coupled with a term representing the Lorentz force. Moreover we assume that both Euler systems are coupled through a quasi-neutrality constraint of the form $n_{i}=n_{e}$. The numerical method which is described in the present document is based on an Asymptotic-Preserving semi-discretization in time of a variant of this two-fluid Euler-Lorentz model with a small perturbation of the quasi-neutrality constraint. Firstly, we present the two-fluid model and the motivations for introducing a small perturbation into the quasi-neutrality equation, then we describe the time semi-discretization of the perturbed model and a fully-discrete finite volume scheme based on it. Finally, we present some numerical results which have been obtained with this method.
Continuous limit of a crowd motion and herding model: Analysis and numerical simulations
Martin Burger, Peter Alexander Markowich and Jan-Frederik Pietschmann
2011, 4(4): 1025-1047 doi: 10.3934/krm.2011.4.1025 +[Abstract](441) +[PDF](3331.3KB)
Abstract:
In this paper we study the continuum limit of a cellular automaton model used for simulating human crowds with herding behaviour. We derive a system of non-linear partial differential equations resembling the Keller-Segel model for chemotaxis, however with a non-monotone interaction. The latter has interesting consequences on the behaviour of the model's solutions, which we highlight in its analysis. In particular we study the possibility of stationary states, the formation of clusters and explore their connection to congestion.
    We also introduce an efficient numerical simulation approach based on an appropriate hybrid discontinuous Galerkin method, which in particular allows flexible treatment of complicated geometries. Extensive numerical studies also provide a better understanding of the strengths and shortcomings of the herding model, in particular we examine trapping effects of crowds behind non-convex obstacles.
Semiclassical limit in a simplified quantum energy-transport model for semiconductors
Li Chen, Xiu-Qing Chen and Ansgar Jüngel
2011, 4(4): 1049-1062 doi: 10.3934/krm.2011.4.1049 +[Abstract](301) +[PDF](387.9KB)
Abstract:
The semiclassical limit in a quantum energy-transport model for semiconductors is proved. The system consists of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; a degenerate elliptic heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The asymptotic limit is based on a priori estimates independent of the scaled Planck constant, obtained from entropy functionals, on the use of Gagliardo-Nirenberg inequalities, and weak compactness methods.
The Spherical Harmonics Expansion model coupled to the Poisson equation
Jan Haskovec, Nader Masmoudi, Christian Schmeiser and Mohamed Lazhar Tayeb
2011, 4(4): 1063-1079 doi: 10.3934/krm.2011.4.1063 +[Abstract](590) +[PDF](241.6KB)
Abstract:
The Spherical Harmonics Expansion (SHE) assumes a momentum distribution function only depending on the microscopic kinetic energy. The SHE-Poisson system describes carrier transport in semiconductors with self-induced electrostatic potential. Existence of weak solutions to the SHE-Poisson system subject to periodic boundary conditions is established, based on appropriate a priori estimates and a Schauder fixed point procedure. The long time behavior of the one-dimensional Dirichlet problem with well prepared boundary data is studied by an entropy-entropy dissipation method. Strong convergence to equilibrium is proven. In contrast to earlier work, the analysis is carried out without the use of the derivation from a kinetic problem.
Averaged kinetic models for flows on unstructured networks
Michael Herty and Christian Ringhofer
2011, 4(4): 1081-1096 doi: 10.3934/krm.2011.4.1081 +[Abstract](305) +[PDF](384.4KB)
Abstract:
We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.
A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping
Shi Jin and Peng Qi
2011, 4(4): 1097-1120 doi: 10.3934/krm.2011.4.1097 +[Abstract](277) +[PDF](1803.7KB)
Abstract:
In this paper, we propose a hybrid method coupling a Schrödinger solver and the Gaussian beam method for the numerical simulation of quantum tunneling through potential barriers or surface hopping across electronic potential energy surfaces. The idea is to use a Schrödinger solver near potential barriers or zones where potential energy surfaces cross, and the Gaussian beam method--which is much more efficient than a direct Schrödinger solver--elsewhere. Buffer zones are used to convert data between the Schrödinger solver and the Gaussian beam solver. Numerical examples show that this method indeed captures quantum tunneling and surface hopping accurately, with a computational cost much lower than a direct quantum solver in the entire domain.
Analysis of a diffusive effective mass model for nanowires
Clément Jourdana and Nicolas Vauchelet
2011, 4(4): 1121-1142 doi: 10.3934/krm.2011.4.1121 +[Abstract](227) +[PDF](471.9KB)
Abstract:
We propose in this paper to derive and analyze a self-consistent model describing the diffusive transport in a nanowire. From a physical point of view, it describes the electron transport in an ultra-scaled confined structure, taking into account the interactions of charged particles with phonons. The transport direction is assumed to be large compared to the wire section and is described by a drift-diffusion equation including effective quantities computed from a Bloch problem in the crystal lattice. The electrostatic potential solves a Poisson equation where the particle density couples on each energy band a two dimensional confinement density with the monodimensional transport density given by the Boltzmann statistics. On the one hand, we study the derivation of this Nanowire Drift-Diffusion Poisson model from a kinetic level description. On the other hand, we present an existence result for this model in a bounded domain.
A problem of moment realizability in quantum statistical physics
Florian Méhats and Olivier Pinaud
2011, 4(4): 1143-1158 doi: 10.3934/krm.2011.4.1143 +[Abstract](249) +[PDF](415.1KB)
Abstract:
This work is a generalization of the results previously obtained in [17] in a one-dimensional setting: we revisit the problem of the minimization of the quantum free energy (entropy + energy) under local constraints (moments) and prove the existence of minimizers in various configurations. While [17] addressed the 1D case on bounded domains, we treat in the present paper the multi-dimensional case as well as unbounded domains and non-linear interactions as Hartree/Hartree-Fock. Moreover, whereas [17] dealt with the first moment only, namely the charge density, we extend the results to the second moment, the current density.
Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport
Stefan Possanner and Claudia Negulescu
2011, 4(4): 1159-1191 doi: 10.3934/krm.2011.4.1159 +[Abstract](381) +[PDF](1110.5KB)
Abstract:
The aim of the present paper is the mathematical study of a linear Boltzmann equation with different matrix collision operators, modelling the spin-polarized, semi-classical electron transport in non-homogeneous ferromagnetic structures. In the collision kernel, the scattering rate is generalized to a hermitian, positive-definite $2\times2$ matrix whose eigenvalues stand for the different scattering rates of, for example, spin-up and spin-down electrons in spintronic applications. We identify four possible structures of linear matrix collision operators that yield existence and uniqueness of a weak solution of the Boltzmann equation for a general Hamilton function. We are able to prove positive-(semi)definiteness of a solution for an operator that features an anti-symmetric structure of the gain respectively the loss term with respect to the occurring matrix products. Furthermore, in order to obtain matrix drift-diffusion equations, we perform the diffusion limit with one of the symmetric operators assuming parabolic spin bands with uniform band gap and in the case that the precession frequency of the spin distribution vector around the exchange field of the Hamiltonian scales with order $\epsilon^2$. Numerical simulations of the here obtained macroscopic model were carried out in non-magnetic/ferromagnetic multilayer structures and for a magnetic Bloch domain wall. The results show that our model can be used to improve the understanding of spin-polarized transport in spintronics applications.

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