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Kinetic & Related Models

August 2018 , Volume 11 , Issue 4

Special issue celebrating the 10th anniversary of KRM

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Preface
Kazuo Aoki, Pierre Degond and Tong Yang
2018, 11(4): i-i doi: 10.3934/krm.201804i +[Abstract](59) +[HTML](41) +[PDF](78.89KB)
Abstract:
A Rosenau-type approach to the approximation of the linear Fokker-Planck equation
Giuseppe Toscani
2018, 11(4): 697-714 doi: 10.3934/krm.2018028 +[Abstract](48) +[HTML](31) +[PDF](389.03KB)
Abstract:

The numerical approximation of the solution of the Fokker-Planck equation is a challenging problem that has been extensively investigated starting from the pioneering paper of Chang and Cooper in 1970 [8]. We revisit this problem at the light of the approximation of the solution to the heat equation proposed by Rosenau [25]. Further, by means of the same idea, we address the problem of a consistent approximation to higher-order linear diffusion equations.

Long time strong convergence to Bose-Einstein distribution for low temperature
Xuguang Lu
2018, 11(4): 715-734 doi: 10.3934/krm.2018029 +[Abstract](47) +[HTML](39) +[PDF](442.97KB)
Abstract:

We study the long time behavior of measure-valued isotropic solutions \begin{document}$F_t$\end{document} of the Boltzmann equation for Bose-Einstein particles for low temperature. The global in time existence of such solutions \begin{document}$F_t$\end{document} that converge at least semi-strongly to equilibrium (the Bose-Einstein distribution) has been proven in previous work and it has been known that the long time strong convergence to equilibrium is equivalent to the long time convergence to the Bose-Einstein condensation. Here we show that if such a solution \begin{document}$F_t$\end{document} as a family of Borel measures satisfies a uniform double-size condition (which is also necessary for the strong convergence), then \begin{document}$F_t$\end{document} converges strongly to equilibrium as \begin{document}$t$\end{document} tends to infinity. We also propose a new condition on the initial datum \begin{document}$F_0$\end{document} such that a corresponding solution \begin{document}$F_t$\end{document} converges strongly to equilibrium.

An asymptotic preserving scheme for kinetic models with singular limit
Alina Chertock, Changhui Tan and Bokai Yan
2018, 11(4): 735-756 doi: 10.3934/krm.2018030 +[Abstract](44) +[HTML](30) +[PDF](792.78KB)
Abstract:

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors, and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.

Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space
Tohru Nakamura, Shinya Nishibata and Naoto Usami
2018, 11(4): 757-793 doi: 10.3934/krm.2018031 +[Abstract](50) +[HTML](33) +[PDF](595.31KB)
Abstract:

In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in one-dimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law
Tohru Nakamura and Shuichi Kawashima
2018, 11(4): 795-819 doi: 10.3934/krm.2018032 +[Abstract](42) +[HTML](34) +[PDF](504.99KB)
Abstract:

In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a $2 × 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.

On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases
Céline Baranger, Marzia Bisi, Stéphane Brull and Laurent Desvillettes
2018, 11(4): 821-858 doi: 10.3934/krm.2018033 +[Abstract](40) +[HTML](30) +[PDF](554.79KB)
Abstract:

In this paper, we propose a formal derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised in [8,16] for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure [6]. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants.

Local sensitivity analysis for the Cucker-Smale model with random inputs
Seung-Yeal Ha and Shi Jin
2018, 11(4): 859-889 doi: 10.3934/krm.2018034 +[Abstract](43) +[HTML](28) +[PDF](559.76KB)
Abstract:

We present pathwise flocking dynamics and local sensitivity analysis for the Cucker-Smale(C-S) model with random communications and initial data. For the deterministic communications, it is well known that the C-S model can model emergent local and global flocking dynamics depending on initial data and integrability of communication function. However, the communication mechanism between agents is not a priori clear and needs to be figured out from observed phenomena and data. Thus, uncertainty in communication is an intrinsic component in the flocking modeling of the C-S model. In this paper, we provide a class of admissible random uncertainties which allows us to perform the local sensitivity analysis for flocking and establish stability to the random C-S model with uncertain communication.

Traveling wave and aggregation in a flux-limited Keller-Segel model
Vincent Calvez, Benoȋt Perthame and Shugo Yasuda
2018, 11(4): 891-909 doi: 10.3934/krm.2018035 +[Abstract](62) +[HTML](88) +[PDF](1212.08KB)
Abstract:

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in population dynamics of chemotactic cells based on the FLKS model with a population growth term.

Our study includes both numerical and theoretical contributions. In the numerical part, we uncover a variety of solution types in the one-dimensional FLKS model additionally to standard Fisher/KPP type traveling wave. The remarkable result is a counter-intuitive backward traveling wave, where the population density initially saturated in a stable state transits toward an unstable state in the local population dynamics. Unexpectedly, we also find that the backward traveling wave solution transits to a localized spiky solution as increasing the stiffness of chemotactic response.

In the theoretical part, we obtain a novel analytic formula for the minimum traveling speed which includes the counter-balancing effect of chemotactic drift vs. reproduction/diffusion in the propagating front. The front propagation speeds of numerical results only slightly deviate from the minimum traveling speeds, except for the localized spiky solutions, even for the backward traveling waves. We also discover an analytic solution of unimodal traveling wave in the large-stiffness limit, which is certainly unstable but exists in a certain range of parameters.

Microscopic solutions of the Boltzmann-Enskog equation in the series representation
Mario Pulvirenti, Sergio Simonella and Anton Trushechkin
2018, 11(4): 911-931 doi: 10.3934/krm.2018036 +[Abstract](37) +[HTML](33) +[PDF](532.64KB)
Abstract:

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel
Philippe Laurençot, Barbara Niethammer and Juan J.L. Velázquez
2018, 11(4): 933-952 doi: 10.3934/krm.2018037 +[Abstract](43) +[HTML](26) +[PDF](454.09KB)
Abstract:

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity \begin{document}$γ < 1$\end{document}. Due to the property of the diagonal kernel, the value of a solution at a given cluster size depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.

On multi-dimensional hypocoercive BGK models
Franz Achleitner, Anton Arnold and Eric A. Carlen
2018, 11(4): 953-1009 doi: 10.3934/krm.2018038 +[Abstract](44) +[HTML](28) +[PDF](861.32KB)
Abstract:

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the $L^1$ distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.

Linear Boltzmann equation and fractional diffusion
Claude Bardos, François Golse and Ivan Moyano
2018, 11(4): 1011-1036 doi: 10.3934/krm.2018039 +[Abstract](44) +[HTML](30) +[PDF](470.71KB)
Abstract:

Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient \begin{document} $\sigma$ \end{document}. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient \begin{document} $\alpha$ \end{document}. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where \begin{document} $\sigma\to+∞$ \end{document} and \begin{document}$ 1-\alpha \sim C/\sigma$ \end{document}, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of \begin{document} $\sqrt{-\Delta}$ \end{document}. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions ("heavy tails") or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes
Weizhu Bao and Chunmei Su
2018, 11(4): 1037-1062 doi: 10.3934/krm.2018040 +[Abstract](51) +[HTML](29) +[PDF](2607.49KB)
Abstract:

We establish a uniform error estimate of a finite difference method for the Klein-Gordon-Schrödinger (KGS) equations with two dimensionless parameters \begin{document}$0<γ≤1$\end{document} and \begin{document}$0<\varepsilon≤1$\end{document}, which are the mass ratio and inversely proportional to the speed of light, respectively. In the simultaneously nonrelativistic and massless limit regimes, i.e., \begin{document}$γ\sim\varepsilon$\end{document} and \begin{document}$\varepsilon \to 0^+$\end{document}, the KGS equations converge singularly to the Schrödinger-Yukawa (SY) equations. When \begin{document}$0<\varepsilon\ll 1$\end{document}, due to the perturbation of the wave operator and/or the incompatibility of the initial data, which is described by two parameters \begin{document}$α≥0$\end{document} and \begin{document}$β≥-1$\end{document}, the solution of the KGS equations oscillates in time with \begin{document}$O(\varepsilon)$\end{document}-wavelength, which requires harsh meshing strategy for classical numerical methods. We propose a uniformly accurate method based on two key points: (ⅰ) reformulating KGS system into an asymptotic consistent formulation, and (ⅱ) applying an integral approximation of the oscillatory term. Using the energy method and the limiting equation via the SY equations with an oscillatory potential, we establish two independent error bounds at \begin{document}$O(h^2+τ^2/\varepsilon)$\end{document} and \begin{document}$O(h^2+τ^2+τ\varepsilon^{α^*}+\varepsilon^{1+α^*})$\end{document} with \begin{document}$h$\end{document} mesh size, \begin{document}$τ$\end{document} time step and \begin{document}$α^* = \min\{1, α, 1+β\}$\end{document}. This implies that the method converges uniformly and optimally with quadratic convergence rate in space and uniformly in time at \begin{document}$O(τ^{4/3})$\end{document} and \begin{document}$O(τ^{1+\frac{α^*}{2+α^*}})$\end{document} for well-prepared (\begin{document}$α^* = 1$\end{document}) and ill-prepared (\begin{document}$0≤α^*<1$\end{document}) initial data, respectively. Thus the \begin{document}$\varepsilon$\end{document}-scalability of the method is \begin{document}$τ = O(1)$\end{document} and \begin{document}$h = O(1)$\end{document} for \begin{document}$0<\varepsilon≤ 1$\end{document}, which is significantly better than classical methods. Numerical results are reported to confirm our error bounds. Finally, the method is applied to study the convergence rates of KGS equations to its limiting models in the simultaneously nonrelativistic and massless limit regimes.

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