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Kinetic & Related Models

2017 , Volume 10 , Issue 4

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Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries
Zbigniew Banach and Wieslaw Larecki
2017, 10(4): 879-900 doi: 10.3934/krm.2017035 +[Abstract](205) +[HTML](8) +[PDF](394.3KB)
The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted \begin{document}$f$\end{document}, must satisfy the inequality \begin{document}$f≤q 1$\end{document} and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the \begin{document}$H$\end{document}-theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.
Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction
Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried and Semjon Vugalter
2017, 10(4): 901-924 doi: 10.3934/krm.2017036 +[Abstract](89) +[HTML](6) +[PDF](502.0KB)
We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.
Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations
Niclas Bernhoff
2017, 10(4): 925-955 doi: 10.3934/krm.2017037 +[Abstract](736) +[HTML](5) +[PDF](593.0KB)
We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting with a Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.
Grossly determined solutions for a Boltzmann-like equation
Thomas Carty
2017, 10(4): 957-976 doi: 10.3934/krm.2017038 +[Abstract](75) +[HTML](8) +[PDF](503.4KB)
In gas dynamics, the connection between the continuum physics model offered by the Navier-Stokes equations and the heat equation and the molecular model offered by the kinetic theory of gases has been understood for some time, especially through the work of Chapman and Enskog, but it has never been established rigorously. This paper established a precise bridge between these two models for a simple linear Boltzman-like equation. Specifically a special class of solutions, the grossly determined solutions, of this kinetic model are shown to exist and satisfy closed form balance equations representing a class of continuum model solutions.
The two dimensional Vlasov-Poisson system with steady spatial asymptotics
Zili Chen, Xiuting Li and Xianwen Zhang
2017, 10(4): 977-1009 doi: 10.3934/krm.2017039 +[Abstract](82) +[HTML](5) +[PDF](518.6KB)
We consider a two dimensional collisionless plasma interacting with a fixed background of positive charge, the density of which depends only upon velocity variable \begin{document}$v$\end{document} and decays as \begin{document}$|v| \to \infty $\end{document}. Suppose that mobile negative ions balance the positive charge as spatial variable \begin{document}$|x|\to \infty $\end{document}, then on the mesoscopic level the system is characterized by the two dimensional Vlasov-Poisson system with steady spatial asymptotics, whose total positive charge and total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior are shown to exist locally in time, and an "almost optimal" criterion for the continuation of these solutions is established.
Cucker-Smale model with normalized communication weights and time delay
Young-Pil Choi and Jan Haskovec
2017, 10(4): 1011-1033 doi: 10.3934/krm.2017040 +[Abstract](151) +[HTML](4) +[PDF](725.4KB)
We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.
Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum
Jishan Fan, Shuxiang Huang and Fucai Li
2017, 10(4): 1035-1053 doi: 10.3934/krm.2017041 +[Abstract](95) +[HTML](7) +[PDF](441.4KB)
This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient \begin{document}$κ(θ)$\end{document} satisfies for some constants \begin{document}$q>0$\end{document}, and \begin{document}$C_1,C_2>0$\end{document}.
The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks
Klemens Fellner, Wolfang Prager and Bao Q. Tang
2017, 10(4): 1055-1087 doi: 10.3934/krm.2017042 +[Abstract](97) +[HTML](5) +[PDF](586.3KB)
In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied.In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition.
   We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients of all but one species are zero.
   For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.
Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system
Hai-Liang Li, Hongjun Yu and Mingying Zhong
2017, 10(4): 1089-1125 doi: 10.3934/krm.2017043 +[Abstract](82) +[HTML](7) +[PDF](645.9KB)
It is interesting to analyze the mutual influence of relativistic effect and electrostatic potential force on the qualitative behaviors of charge particles simulated by the one-species relativistic Vlasov-Poisson-Landau (rVPL) system with the physical Coulombic interaction. In this paper, we first study the spectrum structure on the linearized rVPL system and obtain the optimal time decay rates of the solutions to the linearized system, and then we construct global strong solutions to the nonlinear system around a global relativistic Maxwellian. Finally we make use of time decay rates of the solutions to the linearized system and uniform energy estimates to establish the time decay of the global solution to the original Cauchy problem for the rVPL system to the absolute Maxwellian at the optimal convergence rate \begin{document}$(1+t)^{-3/4}$\end{document}. This time rate is faster than the optimal rate \begin{document}$(1+t)^{-1/4}$\end{document} of classical Vlasov-Poisson-Boltzmann [2,10] and Vlasov-Poisson-Landau system [7,8,17] and this fast time decay rate is caused by the combined influence of relativistic effect and electrostatic potential force.
First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions
Florian Schneider, Andreas Roth and Jochen Kall
2017, 10(4): 1127-1161 doi: 10.3934/krm.2017044 +[Abstract](75) +[HTML](5) +[PDF](13892.5KB)
Mixed-moment models, introduced in [8,44] for one space dimension, are a modification of the method of moments applied to a (linear) kinetic equation, by choosing mixtures of different partial moments. They are well-suited to handle equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimensions. In the context of minimum-entropy models, the resulting hyperbolic system of equations has desirable properties (entropy-diminishing, bounded eigenvalues), removing some drawbacks of the well-known M1 model. We furthermore provide a realizability theory for a first-order system of mixed moments by linking it to the corresponding quarter-moment theory. Additionally, we derive a type of Kershaw closures for mixed-and quarter-moment models, giving an efficient closure (compared to minimum-entropy models). The derived closures are investigated for different benchmark problems.
Diffusive limit with geometric correction of unsteady neutron transport equation
Lei Wu
2017, 10(4): 1163-1203 doi: 10.3934/krm.2017045 +[Abstract](61) +[HTML](4) +[PDF](621.9KB)
We consider the diffusive limit of an unsteady neutron transportequation in a two-dimensional plate with one-speed velocity. We show the solution can be approximated by the sum of interior solution, initial layer, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory in [1] which states the behavior of solution near boundary can be described by the Knudsen layer derived from the Milne problem.
Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three
Zhigang Wu and Weike Wang
2017, 10(4): 1205-1233 doi: 10.3934/krm.2017046 +[Abstract](91) +[HTML](7) +[PDF](559.4KB)
The Cauchy problem of the reduced gravity two and a half layer model in dimension three isconsidered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit two kinds of the generalized Huygens' waves. It is a significant different phenomenon from the Navier-Stokes system. Lastly, as a byproduct, we also extend \begin{document}$L^2(\mathbb{R}^3)$\end{document}-decay rate to \begin{document}$L^p(\mathbb{R}^3)$\end{document}-decay rate with \begin{document}$p>1$\end{document}.
The stability of contact discontinuity for compressible planar magnetohydrodynamics
Haiyan Yin
2017, 10(4): 1235-1253 doi: 10.3934/krm.2017047 +[Abstract](53) +[HTML](6) +[PDF](474.0KB)
This paper is concerned with the planar magnetohydrodynamicswith initial data whose behaviors at far fields \begin{document}$x\rightarrow \pm\infty$\end{document} are different. Motivated by the relationship between planar magnetohydrodynamics and Navier-Stokes, we can prove that the solutions to the planar magnetohydrodynamics tend time-asymptotically to a viscous contact wave which is constructed from a contact discontinuity solution of the Riemann problemon Euler system. This result is proved by the method of elementary energy estimates.
$L^\infty$ resolvent bounds for steady Boltzmann's equation
Kevin Zumbrun
2017, 10(4): 1255-1257 doi: 10.3934/krm.2017048 +[Abstract](72) +[HTML](10) +[PDF](304.2KB)
We derive lower bounds on the resolvent operator for the linearized steady Boltzmann equation over weighted \begin{document}$L^\infty$\end{document}Banach spaces in velocity, comparable to those derived by Pogan & Zumbrun in an analogous weighted \begin{document}$L^2$\end{document} Hilbert space setting.These show in particular that the operator norm of the resolvent kernel is unbounded in \begin{document}$L^p(\mathbb{R})$\end{document} for all \begin{document}$1<p \leq \infty$\end{document}, resolving an apparent discrepancy in behavior between the two settings suggested by previous work.

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