# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

December 2019 , Volume 12 , Issue 6

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Yanbo Hu and
2019, 12(6): 1197-1228 doi: 10.3934/krm.2019046 +[Abstract](379) +[HTML](94) +[PDF](428.29KB)
Abstract:

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

2019, 12(6): 1229-1245 doi: 10.3934/krm.2019047 +[Abstract](353) +[HTML](82) +[PDF](345.66KB)
Abstract:

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.

2019, 12(6): 1247-1271 doi: 10.3934/krm.2019048 +[Abstract](355) +[HTML](86) +[PDF](3318.3KB)
Abstract:

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM, 42(3):934–952, 2004).

2019, 12(6): 1273-1296 doi: 10.3934/krm.2019049 +[Abstract](550) +[HTML](97) +[PDF](11624.51KB)
Abstract:

We consider a kinetic theory approach to model the evacuation of a crowd from bounded domains. The interactions of a person with other pedestrians and the environment, which includes walls, exits, and obstacles, are modeled by using tools of game theory and are transferred to the crowd dynamics. The model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. For the numerical approximation of the solution to our model, we apply an operator splitting scheme which breaks the problem into two pure advection problems and a problem involving the interactions. We compare our numerical results against the data reported in a recent empirical study on evacuation from a room with two exits. For medium and medium-to-large groups of people we achieve good agreement between the computed average people density and flow rate and the respective measured quantities. Through a series of numerical tests we also show that our approach is capable of handling evacuation from a room with one or more exits with variable size, with and without obstacles, and can reproduce lane formation in bidirectional flow in a corridor.

2019, 12(6): 1297-1312 doi: 10.3934/krm.2019050 +[Abstract](320) +[HTML](87) +[PDF](1246.29KB)
Abstract:

We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. We use global bifurcation theory to prove that there is a curve \begin{document}$\mathcal{K}$\end{document} of such states with the following property. The curve begins at the sparking voltage and either the particle density becomes unbounded or the curve ends at the anti-sparking voltage.

2019, 12(6): 1313-1327 doi: 10.3934/krm.2019051 +[Abstract](348) +[HTML](69) +[PDF](353.6KB)
Abstract:

We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small \begin{document}$C^k$\end{document} norms (\begin{document}$k \geq 1$\end{document}) for the charge densities and the electric fields, but attain arbitrarily large \begin{document}$L^\infty$\end{document} norms of them at some later time.

2019, 12(6): 1329-1358 doi: 10.3934/krm.2019052 +[Abstract](362) +[HTML](95) +[PDF](1636.7KB)
Abstract:

We consider a spatially-extended model for a network of interacting FitzHugh-Nagumo neurons without noise, and rigorously establish its mean-field limit towards a nonlocal kinetic equation as the number of neurons goes to infinity. Our approach is based on deterministic methods, and namely on the stability of the solutions of the kinetic equation with respect to their initial data. The main difficulty lies in the adaptation in a deterministic framework of arguments previously introduced for the mean-field limit of stochastic systems of interacting particles with a certain class of locally Lipschitz continuous interaction kernels. This result establishes a rigorous link between the microscopic and mesoscopic scales of observation of the network, which can be further used as an intermediary step to derive macroscopic models. We also propose a numerical scheme for the discretization of the solutions of the kinetic model, based on a particle method, in order to study the dynamics of its solutions, and to compare it with the microscopic model.

2019, 12(6): 1359-1429 doi: 10.3934/krm.2019053 +[Abstract](266) +[HTML](87) +[PDF](694.68KB)
Abstract:

We construct a unique global-in-time solution to the two species Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition, which can be viewed as one of the ideal scattering boundary model. The construction follows a new \begin{document}$L^{2}$\end{document}-\begin{document}$L^{\infty}$\end{document} framework in [3]. In our knowledge this result is the first construction of strong solutions for two species plasma models with self-consistent field in general bounded domains.

2018  Impact Factor: 1.38