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Kinetic & Related Models

October 2018 , Volume 11 , Issue 5

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On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field
Simone Calogero and Stephen Pankavich
2018, 11(5): 1063-1083 doi: 10.3934/krm.2018041 +[Abstract](30) +[HTML](28) +[PDF](427.88KB)

The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global-in-time solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in finite time. Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology.

Boundary layers and stabilization of the singular Keller-Segel system
Hongyun Peng, Zhi-An Wang, Kun Zhao and Changjiang Zhu
2018, 11(5): 1085-1123 doi: 10.3934/krm.2018042 +[Abstract](36) +[HTML](19) +[PDF](751.9KB)

The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε→0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-ε estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L1-estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation
Prasanta Kumar Barik and Ankik Kumar Giri
2018, 11(5): 1125-1138 doi: 10.3934/krm.2018043 +[Abstract](83) +[HTML](24) +[PDF](371.0KB)

In general, the non-conservative approximation of coagulation-fragmentation equations (CFEs) may lead to the occurrence of gelation phenomenon. In this article, it is shown that the non-conservative approximation of CFEs can also provide the existence of mass conserving solutions to CFEs for large classes of unbounded coagulation and fragmentation kernels.

Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling
Liu Liu
2018, 11(5): 1139-1156 doi: 10.3934/krm.2018044 +[Abstract](31) +[HTML](19) +[PDF](492.47KB)

In this paper, we study the generalized polynomial chaos (gPC) based stochastic Galerkin method for the linear semiconductor Boltzmann equation under diffusive scaling and with random inputs from an anisotropic collision kernel and the random initial condition. While the numerical scheme and the proof of uniform-in-Knudsen-number regularity of the distribution function in the random space has been introduced in [15], the main goal of this paper is to first obtain a sharper estimate on the regularity of the solution-an exponential decay towards its local equilibrium, which then lead to the uniform spectral convergence of the stochastic Galerkin method for the problem under study.

Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit
Seung-Yeal Ha, Jeongho Kim and Xiongtao Zhang
2018, 11(5): 1157-1181 doi: 10.3934/krm.2018045 +[Abstract](61) +[HTML](34) +[PDF](532.31KB)
On global solutions to the Vlasov-Poisson system with radiation damping
Meixia Xiao and Xianwen Zhang
2018, 11(5): 1183-1209 doi: 10.3934/krm.2018046 +[Abstract](35) +[HTML](22) +[PDF](550.93KB)

In this paper, the dynamics of three dimensional Vlasov-Poisson system with radiation damping is investigated. We prove global existence of a classical as well as weak solution that propagates boundedness of velocity-space support or velocity-space moment of order two respectively. This kind of solutions possess finite mass but need not necessarily have finite kinetic energy. Moreover, uniqueness of the classical solution is also shown.

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations
Alexander Alekseenko, Truong Nguyen and Aihua Wood
2018, 11(5): 1211-1234 doi: 10.3934/krm.2018047 +[Abstract](35) +[HTML](55) +[PDF](2466.17KB)

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with \begin{document}$O(MN)$\end{document} operations, where \begin{document}$N$\end{document} is the number of the discrete velocity points and \begin{document}$M <N$\end{document}. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations
Yicheng Jiang and Kaijun Zhang
2018, 11(5): 1235-1253 doi: 10.3934/krm.2018048 +[Abstract](35) +[HTML](18) +[PDF](459.2KB)

This paper is concerned with the stability of noncritical/critical traveling waves for nonlocal time-delayed reaction-diffusion equation. When the birth rate function is non-monotone, the solution of the delayed equation is proved to converge time-exponentially to some (monotone or non-monotone) traveling wave profile with wave speed \begin{document}$c>c_*$\end{document}, where \begin{document}$c_*>0$\end{document} is the minimum wave speed, when the initial data is a small perturbation around the wave. However, for the critical traveling waves (\begin{document}$c = c_*$\end{document}), the time-asymptotical stability is only obtained, and the decay rate is not gotten due to some technical restrictions. The proof approach is based on the combination of the anti-weighted method and the nonlinear Halanay inequality but with some new development.

Second-order mixed-moment model with differentiable ansatz function in slab geometry
Florian Schneider
2018, 11(5): 1255-1276 doi: 10.3934/krm.2018049 +[Abstract](24) +[HTML](22) +[PDF](3474.42KB)

Mixed-moment minimum-entropy models (\begin{document}${\rm{M}}{{\rm{M}}_N}$\end{document} models) are known to overcome the zero net-flux problem of full-moment minimum entropy \begin{document}${{\rm{M}}_N}$\end{document} models but lack regularity. We study differentiable mixed-moment models (full zeroth and first moment, half higher moments, called \begin{document}${\rm{DM}}{{\rm{M}}_N}$\end{document} models) for a Fokker-Planck equation in one space dimension. Realizability theory for these modification of mixed moments is derived for second order. Numerical tests are performed with a kinetic first-order finite volume scheme and compared with \begin{document}${{\rm{M}}_N}$\end{document}, classical \begin{document}${\rm{M}}{{\rm{M}}_N}$\end{document} and a \begin{document}${{\rm{P}}_N}$\end{document} reference scheme.

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