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Kinetic & Related Models

April 2018 , Volume 11 , Issue 2

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Entropy production inequalities for the Kac Walk
Eric A. Carlen, Maria C. Carvalho and Amit Einav
2018, 11(2): 219-238 doi: 10.3934/krm.2018012 +[Abstract](496) +[HTML](280) +[PDF](477.57KB)
Abstract:

Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of 'chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the 'Almost' Cercignani Conjecture on the sphere.

Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation
Tomasz Komorowski and Łukasz Stȩpień
2018, 11(2): 239-278 doi: 10.3934/krm.2018013 +[Abstract](404) +[HTML](209) +[PDF](615.36KB)
Abstract:

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion
Maxime Breden
2018, 11(2): 279-301 doi: 10.3934/krm.2018014 +[Abstract](407) +[HTML](192) +[PDF](473.56KB)
Abstract:

In this paper, we investigate the use of so called "duality lemmas" to study the system of discrete coagulation-fragmentation equations with diffusion. When the fragmentation is strong enough with respect to the coagulation, we show that we have creation and propagation of superlinear moments. In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [3], about the regularity of the solutions in the pure coagulation case, to strong fragmentation models.

A Vlasov-Poisson plasma of infinite mass with a point charge
Gang Li and Xianwen Zhang
2018, 11(2): 303-336 doi: 10.3934/krm.2018015 +[Abstract](515) +[HTML](283) +[PDF](511.26KB)
Abstract:

We study the time evolution of the three dimensional Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass. We prove the existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space, but not integrable. This result extends a previous theorem for Yukawa potential obtained in [10] to the case of Coulomb interaction.

On a Fokker-Planck equation for wealth distribution
Marco Torregrossa and Giuseppe Toscani
2018, 11(2): 337-355 doi: 10.3934/krm.2018016 +[Abstract](482) +[HTML](260) +[PDF](477.51KB)
Abstract:

We study here a Fokker-Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society [2, 10, 22]. In particular, we analyze the large-time behavior of the solution, by showing that convergence to the steady state can be obtained in various norms at different rates.

Invariant measures for a stochastic Fokker-Planck equation
Sylvain De Moor, Luis Miguel Rodrigues and Julien Vovelle
2018, 11(2): 357-395 doi: 10.3934/krm.2018017 +[Abstract](422) +[HTML](185) +[PDF](641.91KB)
Abstract:

We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term. When the noise intensity is not too large, we solve the corresponding Cauchy problem in a space of functions ensuring good localization in the velocity variable. Then we show under similar conditions that the generated dynamics, with prescribed total mass, admits a unique invariant measure which is exponentially mixing. The proof relies on hypocoercive estimates and hypoelliptic regularity. At last we provide an explicit example showing that our analytic framework does require some smallness condition on the noise intensity.

Regularity theorems for a biological network formulation model in two space dimensions
Xiangsheng Xu
2018, 11(2): 397-408 doi: 10.3934/krm.2018018 +[Abstract](395) +[HTML](219) +[PDF](366.63KB)
Abstract:

We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [13]. A consequence of these results is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.

Numerical schemes for kinetic equation with diffusion limit and anomalous time scale
Hélène Hivert
2018, 11(2): 409-439 doi: 10.3934/krm.2018019 +[Abstract](592) +[HTML](294) +[PDF](718.78KB)
Abstract:

In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form \begin{document}${\varepsilon ^2}\left( {1 + \left| {\ln \left( \varepsilon \right)} \right|} \right)$\end{document}. When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. This work deals with a critical exponent between these two cases, for which an anomalous time scale must be used to find a standard diffusion limit. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degeneracy \begin{document}$ \varepsilon\to0$\end{document} makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Moreover, we prove that the convergence towards the diffusion limit happens with two scales, the second being very slow. The schemes we propose are designed to respect this asymptotic behavior. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

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