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Kinetic & Related Models

January 2021 , Volume 14 , Issue 1

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Kinetic modelling of colonies of myxobacteria
Sabine Hittmeir, Laura Kanzler, Angelika Manhart and Christian Schmeiser
2021, 14(1): 1-24 doi: 10.3934/krm.2020046 +[Abstract](455) +[HTML](137) +[PDF](1569.69KB)

A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions.

For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented.

BGK model of the multi-species Uehling-Uhlenbeck equation
Gi-Chan Bae, Christian Klingenberg, Marlies Pirner and Seok-Bae Yun
2021, 14(1): 25-44 doi: 10.3934/krm.2020047 +[Abstract](431) +[HTML](148) +[PDF](356.64KB)

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and \begin{document}$ H $\end{document}-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficients of the local equilibriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.

Opinion formation systems via deterministic particles approximation
Simone Fagioli and Emanuela Radici
2021, 14(1): 45-76 doi: 10.3934/krm.2020048 +[Abstract](389) +[HTML](132) +[PDF](3930.95KB)

We propose an ODE-based derivation for a generalized class of opinion formation models either for single and multiple species (followers, leaders, trolls). The approach is purely deterministic and the evolution of the single opinion is determined by the competition between two mechanisms: the opinion diffusion and the compromise process. Such deterministic approach allows to recover in the limit an aggregation/(nonlinear)diffusion system of PDEs for the macroscopic opinion densities.

On two properties of the Fisher information
Nicolas Rougerie
2021, 14(1): 77-88 doi: 10.3934/krm.2020049 +[Abstract](297) +[HTML](85) +[PDF](330.03KB)

Alternative proofs for the superadditivity and the affinity (in the large system limit) of the usual and some fractional Fisher informations of a probability density of many variables are provided. They are consequences of the fact that such informations can be interpreted as quantum kinetic energies.

Superposition principle and schemes for measure differential equations
Fabio Camilli, Giulia Cavagnari, Raul De Maio and Benedetto Piccoli
2021, 14(1): 89-113 doi: 10.3934/krm.2020050 +[Abstract](368) +[HTML](83) +[PDF](557.51KB)

Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savaré, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.

Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations
Juan Pablo Pinasco, Mauro Rodriguez Cartabia and Nicolas Saintier
2021, 14(1): 115-148 doi: 10.3934/krm.2020051 +[Abstract](327) +[HTML](84) +[PDF](804.22KB)

In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases by a small amount \begin{document}$ h $\end{document} the probability of playing the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when \begin{document}$ h $\end{document} goes to zero.

We study the relationship between this equation and the well known replicator equations, showing the equivalence between the concepts of Nash equilibria, stationary solutions of the partial differential equation, and the equilibria of the replicator equations. Finally, we relate the long-time behavior of solutions to the partial differential equation and the stability of the replicator equations.

Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem
Mehdi Badsi
2021, 14(1): 149-174 doi: 10.3934/krm.2020052 +[Abstract](387) +[HTML](102) +[PDF](429.9KB)

The mathematical description of the interaction between a collisional plasma and an absorbing wall is a challenging issue. In this paper, we propose to model this interaction by considering a stationary bi-species Vlasov-Poisson-Boltzmann boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be uniquely determined from the ambipolarity of the particles flows as the unique solution of a nonlinear equation. We also prove that it is an increasing function of the electrons re-emission coefficient at the wall. Based on the Schauder fixed point theorem, our analysis establishes the existence of a solution provided, on the one hand that the incoming ions density satisfies a moment condition that generalizes the Historical Bohm criterion, and on the other hand that the collision frequency does not exceed a critical value whose definition is subordinated to the strict validity of our generalized Bohm criterion.

Navier-Stokes limit of globally hyperbolic moment equations
Zhiting Ma
2021, 14(1): 175-197 doi: 10.3934/krm.2021001 +[Abstract](263) +[HTML](37) +[PDF](365.26KB)

This paper is concerned with the Navier-Stokes limit of a class of globally hyperbolic moment equations from the Boltzmann equation. we show that the Navier-Stokes equations can be formally derived from the hyperbolic moment equations for various different collision mechanisms. Furthermore, the formal limit is justified rigorously by using an energy method. It should be noted that the hyperbolic moment equations are in non-conservative form and do not have a convex entropy function, therefore some additional efforts are required in the justification.

2019  Impact Factor: 1.311




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