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Kinetic & Related Models

September 2017 , Volume 10 , Issue 3

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Fractional diffusion limit of a linear kinetic equation in a bounded domain
Pedro Aceves-Sánchez and Christian Schmeiser
2017, 10(3): 541-551 doi: 10.3934/krm.2017021 +[Abstract](671) +[HTML](6) +[PDF](360.5KB)
Abstract:

A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.

Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime
Weizhu Bao, Loïc Le Treust and Florian Méhats
2017, 10(3): 553-571 doi: 10.3934/krm.2017022 +[Abstract](482) +[HTML](1) +[PDF](476.0KB)
Abstract:

We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.

Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules
Alexander V. Bobylev and Irene M. Gamba
2017, 10(3): 573-585 doi: 10.3934/krm.2017023 +[Abstract](401) +[HTML](3) +[PDF](409.9KB)
Abstract:

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models
María J. Cáceres and Ricarda Schneider
2017, 10(3): 587-612 doi: 10.3934/krm.2017024 +[Abstract](530) +[HTML](5) +[PDF](696.2KB)
Abstract:

Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are often used to describe neural networks. Recently, new mathematical results have provided a better understanding of them. It has been proved that a fully excitatory network can blow-up in finite time, while a fully inhibitory network has a global in time solution for any initial data. A general description of the steady states of a purely excitatory or inhibitory network has been also given. We extend this study to the system composed of an excitatory population and an inhibitory one. We prove that this system can also blow-up in finite time and analyse its steady states and long time behaviour. Besides, we illustrate our analytical description with some numerical results. The main tools used to reach our aims are: the control of an exponential moment for the blow-up results, a more complicate strategy than that considered in [5] for studying the number of steady states, entropy methods combined with Poincaré inequalities for the long time behaviour and, finally, high order numerical schemes together with parallel computation techniques in order to obtain our numerical results.

Numerical study of a particle method for gradient flows
José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini and Gershon Wolansky
2017, 10(3): 613-641 doi: 10.3934/krm.2017025 +[Abstract](713) +[HTML](37) +[PDF](728.3KB)
Abstract:

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.

Asymptotic preserving and time diminishing schemes for rarefied gas dynamic
Nicolas Crouseilles, Giacomo Dimarco and Mohammed Lemou
2017, 10(3): 643-668 doi: 10.3934/krm.2017026 +[Abstract](414) +[HTML](3) +[PDF](1084.1KB)
Abstract:

In this work, we introduce a new class of numerical schemes for rarefied gas dynamic problems described by collisional kinetic equations. The idea consists in reformulating the problem using a micro-macro decomposition and successively in solving the microscopic part by using asymptotic preserving Monte Carlo methods. We consider two types of decompositions, the first leading to the Euler system of gas dynamics while the second to the Navier-Stokes equations for the macroscopic part. In addition, the particle method which solves the microscopic part is designed in such a way that the global scheme becomes computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error due to the particle part of the solution decreases as the system approach the equilibrium state. This causes the method to degenerate to the sole solution of the macroscopic hydrodynamic equations (Euler or Navier-Stokes) in the limit of infinite number of collisions. In a last part, we will show the behaviors of this new approach in comparisons to standard Monte Carlo techniques for solving the kinetic equation by testing it on different problems which typically arise in rarefied gas dynamic simulations.

Finite range method of approximation for balance laws in measure spaces
Piotr Gwiazda, Piotr Orlinski and Agnieszka Ulikowska
2017, 10(3): 669-688 doi: 10.3934/krm.2017027 +[Abstract](497) +[HTML](2) +[PDF](574.6KB)
Abstract:

In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

Emergent dynamics in the interactions of Cucker-Smale ensembles
Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang and Xiongtao Zhang
2017, 10(3): 689-723 doi: 10.3934/krm.2017028 +[Abstract](467) +[HTML](4) +[PDF](595.6KB)
Abstract:

Merging and separation of flocking groups are often observed in our natural complex systems. In this paper, we employ the Cucker-Smale particle model to model such merging and separation phenomena. For definiteness, we consider the interaction of two homogeneous Cucker-Smale ensembles and present several sufficient frameworks for mono-cluster flocking, bi-cluster flocking and partial flocking in terms of coupling strength, communication weight, and initial configurations.

Fractional kinetic hierarchies and intermittency
Anatoly N. Kochubei and Yuri Kondratiev
2017, 10(3): 725-740 doi: 10.3934/krm.2017029 +[Abstract](501) +[HTML](1) +[PDF](393.5KB)
Abstract:

We consider general convolutional derivatives and related fractional statistical dynamics of continuous interacting particle systems. We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. Conditions for the intermittency property of fractional kinetic dynamics are obtained.

Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces
Fucai Li, Yanmin Mu and Dehua Wang
2017, 10(3): 741-784 doi: 10.3934/krm.2017030 +[Abstract](531) +[HTML](5) +[PDF](668.3KB)
Abstract:

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.

Decay property for solutions to plate type equations with variable coefficients
Shikuan Mao and Yongqin Liu
2017, 10(3): 785-797 doi: 10.3934/krm.2017031 +[Abstract](452) +[HTML](1) +[PDF](404.5KB)
Abstract:

In this paper we consider the initial value problem of a rotational inertial model for plate type equations with variable coefficients and memory in \begin{document}$\mathbb{R}^n\ (n≥q1)$\end{document}. We study the decay and the regularity-loss property for this equation in the spirit of [12,15], and characterize the decay and regularity property by a function in the spectral space.

On a linear runs and tumbles equation
Stéphane Mischler and Qilong Weng
2017, 10(3): 799-822 doi: 10.3934/krm.2017032 +[Abstract](414) +[HTML](2) +[PDF](513.6KB)
Abstract:

We consider a linear runs and tumbles equation in dimension \begin{document}$d ≥ 1$\end{document} for which we establish the existence of a unique positive and normalized steady state as well as its asymptotic stability, improving similar results obtained by Calvez et al. [8] in dimension \begin{document}$d=1$\end{document}. Our analysis is based on the Krein-Rutman theory revisited in [23] together with some new moment estimates for proving confinement mechanism as well as dispersion, multiplicator and averaging lemma arguments for proving some regularity property of suitable iterated averaging quantities.

Kinetic models for traffic flow resulting in a reduced space of microscopic velocities
Gabriella Puppo, Matteo Semplice, Andrea Tosin and Giuseppe Visconti
2017, 10(3): 823-854 doi: 10.3934/krm.2017033 +[Abstract](467) +[HTML](7) +[PDF](1674.7KB)
Abstract:

The purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based on a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes. Moreover, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic equations using the closure provided by the kinetic model.

Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source
Shijie Shi, Zhengrong Liu and Hai-Yang Jin
2017, 10(3): 855-878 doi: 10.3934/krm.2017034 +[Abstract](708) +[HTML](6) +[PDF](564.5KB)
Abstract:

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source

in a smooth bounded domain \begin{document}$Ω \subset \mathbb{R}^n(n≥ 1)$\end{document}, with homogeneous Neumann boundary conditions and nonnegative initial data \begin{document}$(u_0,v_0,w_0)$\end{document} satisfying suitable regularity, where \begin{document}$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$\end{document} and \begin{document}$f$\end{document} is a smooth growth source satisfying \begin{document}$f(0)≥ 0$\end{document} and

When \begin{document}$χα=ξγ$\end{document} (i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter \begin{document}$θ$\end{document} and the space dimension \begin{document}$n$\end{document} satisfy

Furthermore, when \begin{document}$f(u)=μ u(1-u)$\end{document} and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if \begin{document}$μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$\end{document}, the solution \begin{document}$(u,v,w)$\end{document} exponentially stabilizes to the constant stationary solution \begin{document}$(1,\frac{α}{β},\frac{γ}{δ})$\end{document} in the case of \begin{document}$1≤ n≤ 9$\end{document}. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.

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