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Kinetic & Related Models

April 2019 , Volume 12 , Issue 2

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Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D
Maxime Hauray and Samir Salem
2019, 12(2): 269-302 doi: 10.3934/krm.2019012 +[Abstract](117) +[HTML](68) +[PDF](617.67KB)

We consider a particle system in 1D, interacting via repulsive or attractive Coulomb forces. We prove the trajectorial propagation of molecular chaos towards a nonlinear SDE associated to the Vlasov-Poisson-Fokker-Planck equation. We obtain a quantitative estimate of convergence in the mean in MKW metric of order one, with an optimal convergence rate of order $N^{-1/2}$. We also prove some exponential concentration inequalities of the associated empirical measures. A key argument is a weak-strong stability estimate on the (nonlinear) VPFP equation, that we are able to adapt for the particle system in some sense.

Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion
Marco Di Francesco and Yahya Jaafra
2019, 12(2): 303-322 doi: 10.3934/krm.2019013 +[Abstract](116) +[HTML](61) +[PDF](700.09KB)

In this paper we consider a one-dimensional nonlocal interaction equation with quadratic porous-medium type diffusion in which the interaction kernels are attractive, nonnegative, and integrable on the real line. Earlier results in the literature have shown existence of nontrivial steady states if the $L^1$ norm of the kernel $G$ is larger than the diffusion constant $\varepsilon$. In this paper we aim at showing that this equation exhibits a 'multiple' behavior, in that solutions can either converge to the nontrivial steady states or decay to zero for large times. We prove the former situation holds in case the initial conditions are concentrated enough and 'close' to the steady state in the $∞$-Wasserstein distance. Moreover, we prove that solutions decay to zero for large times in the diffusion-dominated regime $\varepsilon≥ \|G\|_{L^1}$. Finally, we show two partial results suggesting that the large-time decay also holds in the complementary regime $\varepsilon < \|G\|_{L^1}$ for initial data with large enough second moment. We use numerical simulations both to validate our local asymptotic stability result and to support our conjecture on the large time decay.

On a Boltzmann equation for Haldane statistics
Leif Arkeryd and Anne Nouri
2019, 12(2): 323-346 doi: 10.3934/krm.2019014 +[Abstract](106) +[HTML](67) +[PDF](446.56KB)

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper Haldane [10] proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong $L^1$ solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stabililty. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

Non-uniqueness of weak solutions of the Quantum-Hydrodynamic system
Peter Markowich and Jesús Sierra
2019, 12(2): 347-356 doi: 10.3934/krm.2019015 +[Abstract](85) +[HTML](66) +[PDF](352.77KB)

We investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization.

A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data
Bingkang Huang and Lan Zhang
2019, 12(2): 357-396 doi: 10.3934/krm.2019016 +[Abstract](112) +[HTML](88) +[PDF](561.64KB)

We present a two-dimensional coupled system for particles and compressible conducting fluid in an electromagnetic field interactions, which the kinetic Vlasov-Fokker-Planck model for particle part and the isentropic compressible MHD equations for the fluid part, respectively, and these separate systems are coupled with the drag force. For this specific coupled system, a sufficient framework for the global existence of classical solutions with large initial data which may contain vacuum is established.

On the interplay between behavioral dynamics and social interactions in human crowds
Nicola Bellomo, Livio Gibelli and Nisrine Outada
2019, 12(2): 397-409 doi: 10.3934/krm.2019017 +[Abstract](102) +[HTML](57) +[PDF](656.01KB)

This paper presents a computational modeling approach to the dynamics of human crowds, where social interactions can have an important influence on the behavioral dynamics of pedestrians. The modeling of the contagion and propagation of emotional states is carried out by looking at real physical situations where safety problems might arise in some specific circumstances. The approach is based on the methods of the kinetic theory of active particles. The evacuation of a metro station is simulated to enlighten the role of the emotional state in the overall dynamics.

Emergence of aggregation in the swarm sphere model with adaptive coupling laws
Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee and Se Eun Noh
2019, 12(2): 411-444 doi: 10.3934/krm.2019018 +[Abstract](109) +[HTML](84) +[PDF](561.94KB)

We present aggregation estimates for the swarm sphere model equipped with the adaptive coupling laws on a sphere. The temporal evolution of coupling strength is determined by a feedback rule incorporating the balance between relative spatial variations and linear damping. For the analytical treatment, we employ two adaptive feedback laws, namely anti-Hebbian and Hebbian laws. For the anti-Hebbian law, we provide a sufficient framework leading to the complete aggregation in which all particles aggregate to the same position and behave like one big point cluster asymptotically. Our frameworks are given in terms of the initial positions and the coupling strengths. For the Hebbian law, we provide proper subsets of the basin of attractions for the complete aggregation and bi-polar aggregation where particles aggregate to the north pole and south pole simultaneously. We also present a uniform \begin{document}$\ell_p$\end{document}-stability of the swarm sphere model with an adaptive coupling with respect to the initial data when the complete aggregation occurs exponentially fast.

Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness
Marcel Braukhoff
2019, 12(2): 445-482 doi: 10.3934/krm.2019019 +[Abstract](93) +[HTML](60) +[PDF](575.17KB)

A semiconductor Boltzmann equation with a non-linear BGK-type collision operator is analyzed for a cloud of ultracold atoms in an optical lattice:

This system contains an interaction potential \begin{document}$n_f(x,t): = ∈t_{\mathbb{T}^d}f(x,p,t)dp$\end{document} being significantly more singular than the Coulomb potential, which is used in the Vlasov-Poisson system. This causes major structural difficulties in the analysis. Furthermore, \begin{document}$ε(p) = -\sum_{i = 1}^d$\end{document} \begin{document}$\cos(2π p_i)$\end{document} is the dispersion relation and \begin{document}$\mathcal{F}_f$\end{document} denotes the Fermi-Dirac equilibrium distribution, which depends non-linearly on \begin{document}$f$\end{document} in this context.

In a dilute plasma—without collisions (r.h.s\begin{document}$. = 0$\end{document})—this system is closely related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial data that the semiconductor Boltzmann equation possesses a local, analytic solution. Here, we exploit the techniques of Mouhout and Villani by using Gevrey-type norms which vary over time. In addition, it is proved that this equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac equilibrium distribution functions.

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