Showing papers in "Kinetic and Related Models in 2009"

Double milling in self-propelled swarms from kinetic theory

Abstract: We present a kinetic theory for swarming systems of interacting, self-propelled discrete particles. Starting from the Liouville equation for the many-body problem we derive a kinetic equation for the single particle probability distribution function and the related macroscopic hydrodynamic equations. General solutions include flocks of constant density and fixed velocity and other non-trivial morphologies such as compactly supported rotating mills. The kinetic theory approach leads us to the identification of macroscopic structures otherwise not recognized as solutions of the hydrodynamic equations, such as double mills of two superimposed flows. We find the conditions allowing for the existence of such solutions and compare to the case of single mills.

A review of boltzmann equation with singular kernels

Abstract: We review recent results about Boltzmann equation for singular or non cutoff cross-sections. Both spatially homogeneous and inhomogeneous Boltzmann equations are considered, and ideas related to Landau equation are explained. Various technical tools are presented, together with applications to existence and regularization issues.

Semilagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics

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.

Local Hilbert expansion for the Boltzmann equation

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.

A Boltzmann-like equation for choice formation

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.

Orientation Waves in a Director Field with Rotational Inertia

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.

Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution

Abstract: A finite Larmor radius approximation is rigourously derived from the Vlasov equation, in the limit of large (and uniform) external magnetic field. Existence and uniqueness of a solution is proven in the stationary frame.

Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHDmodel

Abstract: We prove some regularity conditions for the MHD equations with partial viscous terms and the Leray-$\alpha$-MHD model. Since the solutions to the Leray-$\alpha$-MHD model are smoother than that of the original MHD equations, we are able to obtain better regularity conditions in terms of the magnetic field $B$ only.

The multi-water-bag equations for collisionless kinetic modeling

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.

The quasi-invariant limit for a kinetic model of sociological collective behavior

Abstract: The paper is devoted to the study of the asymptotic behavior of a kinetic model proposed to forecast the phenomenon of opinion formation, with both effect of self-thinking and compromise between individuals. By supposing that the effects of self-thinking and compromise are very weak, we asymptotically deduce some simpler models who loose the kinetic structure. We explicitly characterize the asymptotic state of the limiting equation and study the speed of convergence towards equilibrium.

Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

Abstract: In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time This is a Gevrey regularizing effect for non smooth initial datum The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier

On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Abstract: We consider a 1D Schrodinger 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 symmetrization of the relativistic Euler equations with severalspatial variables

Abstract: We consider the Euler equations governing relativistic compressible fluids evolving in the Minkowski spacetime with several spatial variables. We propose a new symmetrization which makes sense for solutions containing vacuum states and, for instance, applies to the case of compactly supported solutions which are important to model star dynamics. Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold and remains bounded away from the light speed, we deduce a local-in-time existence result for solutions containing vacuum states. We also observe that the support of compactly supported solutions does not expand as time evolves.

On long-time behavior of monocharged and neutral plasma in one and one-half dimensions

Abstract: The motion of a collisionless plasma - a high-temperature, low-density, ionized gas - is described by the Vlasov-Maxwell system. In the presence of large velocities, relativistic corrections are meaningful, and when symmetry of the particle densities is assumed this formally becomes the relativistic Vlasov-Poisson system. These equations are considered in one space dimension and two momentum dimensions in both the monocharged (i.e., single species of ion) and neutral cases. The behavior of solutions to these systems is studied for large times, yielding estimates on the growth of particle momenta and a lower bound, uniform-in-time, on norms of the charge density. We also present similar results in the same dimensional settings for the classical Vlasov-Poisson system, which excludes relativistic effects.

A smooth model for fiber lay-down processes and its diffusion approximations

Abstract: In this paper we improve and investigate a stochastic model and its associated Fokker-Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials which has been developed in [2]. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. In the present paper we remove an obvious drawback of the model, namely the non-differentiability of the paths of the process. We develop a model with smoother trajectories and investigate the relations between the different models looking at different scalings and diffusion approximations. Moreover, we compare the numerical results to simulations of the full physical process.

Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel

Abstract: This paper is devoted to numerical simulation of a charged particle beam submitted to a strong oscillating electric field. For that, we consider a two-scale numerical approach as follows: we first recall the two-scale model which is obtained by using two-scale convergence techniques; then, we numerically solve this limit model by using a backward semi-lagrangian method and we propose a new mesh of the phase space which allows us to simplify the solution of the Poisson's equation. Finally, we present some numerical results which have been obtained by the new method, and we validate its efficiency through long time simulations.

Three-dimensional instabilities in non-parallel shear stratified flows

Abstract: The instabilities of non-parallel flows ($\overline{U}(x_3)$, $\overline{V}(x_3), 0)$ ($\overline{V} e 0$) 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.

Existence and sharp localization in velocity of small-amplitude Boltzmann shocks

Abstract: Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman-Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere 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 square-root Maxwellian weighted norm for which it is self-adjoint, 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 near-Maxwellian rate, rather than the square-root rate obtained in previous analyses.

Particle approximation of some Landau equations

Abstract: We consider a class of nonlinear partial-differential equations, including the spatially homogeneous Fokker-Planck-Landau equation for Maxwell (or pseudo-Maxwell) molecules. Continuing the work of [6, 7, 4], we propose a probabilistic interpretation of such a P.D.E. in terms of a nonlinear stochastic differential equation driven by a standard Brownian motion. We derive a numerical scheme, based on a system of $n$ particles driven by $n$ Brownian motions, and study its rate of convergence. We finally deal with the possible extension of our numerical scheme to the case of the Landau equation for soft potentials, and give some numerical results.

Stability of the travelling wave in a 2D weakly nonlinear Stefan problem

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.

Nonlinear stability of boundary layer solutions for generalized benjamin-bona-mahony-burgers equation in the half space

Abstract: This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $ R_+$ $u_t-$u txx -u xx $+f(u)_{x}=0,\ \ \ \ \ t>0,\ \ x\in R_+, $ $u(0,x)=u_0(x)\to u_+,\ \ \ as \ \ x\to +\infty,$ $u(t,0)=u_b$. Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+$≠$u_b$ are two given constant states and the nonlinear function $f(u)$ is a general smooth function.   Asymptotic stability and convergence rates (both algebraic and exponential) of global solution $u(t,x)$ to the above initial-boundary value problem toward the boundary layer solution $\phi(x)$ are established in [9] for both the non-degenerate case $f'(u_+)<0$ and the degenerate case $f'(u_+)=0$. We note, however, that the analysis in [9] relies heavily on the assumption that $f(u)$ is strictly convex. Moreover, for the non-degenerate case, if the boundary layer solution $\phi(x)$ is monotonically decreasing, only the stability of weak boundary layer solution is obtained in [9]. This manuscript is concerned with the non-degenerate case and our main purpose is two-fold: Firstly, for general smooth nonlinear function $f(u)$, we study the global stability of weak boundary layer solutions to the above initial-boundary value problem. Secondly, when $f(u)$ is convex and the corresponding boundary layer solutions are monotonically decreasing, we discuss the local nonlinear stability of strong boundary layer solution. In both cases, the corresponding decay rates are also obtained.

Modelling and simulation of a solar updraft tower

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.

1D numerical simulation of the mep mathematical model in ballistic diode problem

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.

Derivation of a two-fluids model for a Bose gas from a quantum kinetic system

Abstract: We formally derive the hydrodynamic limit of a system modelling a bosons gas having a condensed part, made of a quantum kinetic and a Gross-Pitaevskii equation. The limit model, which is a two-fluids Euler system, is approximated by an isentropic system, which is then studied. We find in particular some conditions for the hyperbolicity, and we study the weak solutions. A numerical example is given at the end.

A note on the time decay of solutions for the linearized Wigner-Poisson system

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. Dedicated to the memory of Marcello Anile

Milne problem for non-grey radiative transfer

Abstract: We study the non-linear Milne problem for radiative transfer equation: after proving the existence of a brightness temperature, we propose and evaluate different formulas for evaluating it. Numerical tests show that as much as 20% difference between surface temperature and brightness temperature may be exhibited. An analytical expression for the out-coming flux is also given.

$\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases

Abstract: The trend to equilibrium of a quaternary mixture of monatomic gases undergoing a reversible reaction of bimolecular type is studied in a quite rigorous mathematical picture within the framework of Boltzmann equation extended to chemically reacting mixtures of gases The $\mathcal H$-theorem and entropy inequality allow to prove two main results under the assumption of uniformly boundedness and equicontinuity of the distribution functions One of the results establishes the tendency of a reacting mixture to evolve to an equilibrium state as time becomes large The other states that the solution of the Boltzmann equation for chemically reacting mixtures of gases converges in strong $L^1$-sense to its equilibrium solution

Variational characterizations of the effective multiplication factor of a nuclear reactor core

Abstract: We prove some inf--sup and sup--inf formulae for the so--called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy--dependent diffusion equation.

A local existence result for a plasma physics modelcontaining a fully coupled magnetic field

Abstract: A local existence theorem is proved for classical solutions of the Vlasov-Poisswell system, a set of collisionless equations used in plasma physics Although the method employed is standard, there are several technical difficulties in the treatment of this system that arise mainly from the, compared to related systems, special form of the electric-field term Furthermore, uniqueness of classical solutions is proved and a continuation criterion for solutions well known for other collisionless kinetic equations is established Finally, a global existence result for a regularized version of the system is derived and comments are given on the problem of obtaining global weak solutions