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Kinetic & Related Models

2009 , Volume 2 , Issue 3

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Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation
Naoufel Ben Abdallah and  Hédia Chaker
2009, 2(3): 403-424 doi: 10.3934/krm.2009.2.403 +[Abstract](41) +[PDF](263.3KB)
Abstract:
In a previous work [J. of Hyperbolic Diff. Eq. 4, pp. 679-704 (2007)], the high field asymptotics of the fermionic Boltzmann equation has been proven to lead to a nonlinear conservation law for the particle density. Under symmetry conditions on the collission cross section, the nonlinear limiting flux is parallel to the force field. In the present work, we investigate the orthogonal direction to the electric field, and prove after a suitable rescaling that the behaviour is governed by the original conservation law with an additional nonlinear diffusion in the orthogonal (to the force field) direction. The main tool used in the convergence proof is a new estimate for the dissipation of the family of entropies introduced in the above cited work. While the entropy dissipation is usually estimated by quantities of the type dist $(f, F_{eq})$ representing the distance of the distribution function to the equilibrium set, the new estimate involves a quantity of the form $\int$ dist $(f, F_{eq}(\u)) d\mu(\u)$, where the macroscopic equilibria depend on a velocity variable $\u$ and $\mu$ is a probability measure. This estimate allows to control high velocities, pass to the limit in the diffusion current and shows the convergence to the entropy solution of the limiting equation.
Identification of photon sources, stochastically embedded in an interstellar cloud
S. Aiello , Luigi Barletti and  Aldo Belleni-Morante
2009, 2(3): 425-432 doi: 10.3934/krm.2009.2.425 +[Abstract](51) +[PDF](130.1KB)
Abstract:
Photon transport is considered in an interstellar cloud containing one or several photon sources (stars), defined by $q_i\delta( x- x_{\i})\,i=1,2,\ldots,$ where the locations $x_i$'s are given in a stochastic way. First, the case is examined of a single source of intensity $q_1$ and located at $x_1$ with a probability density $p_1 = \p(x_1)$, such that $\p(x_1)\geq 0$ and $\int_V \p(x_1)\dx_1 = 1$, where $V \subset \R^3$ is the region occupied by the cloud. Then, a Boltzmann-like equation for the average photon distribution function < n >$(x,u;x_1)$ is derived and it is shown that $\p(x_1)$ can be evaluated starting from a far-field measurement of < n >. Finally, the case of two or more photon sources is discussed: the corresponding results are reasonably simple if $\p(x_1,x_2) = \p_1(x_1)\p_2(x_2)$, i.e. if the locations of the two photon source are "independent".
The quasi-invariant limit for a kinetic model of sociological collective behavior
Laurent Boudin and  Francesco Salvarani
2009, 2(3): 433-449 doi: 10.3934/krm.2009.2.433 +[Abstract](48) +[PDF](299.5KB)
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.
Particle approximation of some Landau equations
Nicolas Fournier
2009, 2(3): 451-464 doi: 10.3934/krm.2009.2.451 +[Abstract](42) +[PDF](245.4KB)
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.
On long-time behavior of monocharged and neutral plasma in one and one-half dimensions
Robert Glassey , Stephen Pankavich and  Jack Schaeffer
2009, 2(3): 465-488 doi: 10.3934/krm.2009.2.465 +[Abstract](29) +[PDF](246.3KB)
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
Michael Herty , Axel Klar , Sébastien Motsch and  Ferdinand Olawsky
2009, 2(3): 489-502 doi: 10.3934/krm.2009.2.489 +[Abstract](46) +[PDF](766.4KB)
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.
A local existence result for a plasma physics model containing a fully coupled magnetic field
Martin Seehafer
2009, 2(3): 503-520 doi: 10.3934/krm.2009.2.503 +[Abstract](31) +[PDF](222.4KB)
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.
Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space
Hui Yin and  Huijiang Zhao
2009, 2(3): 521-550 doi: 10.3934/krm.2009.2.521 +[Abstract](39) +[PDF](331.1KB)
Abstract:
This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $ R_+$

$u_t-$utxx-uxx$+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.

2016  Impact Factor: 1.261

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