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Kinetic & Related Models

2018 , Volume 11 , Issue 3

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Wall effect on the motion of a rigid body immersed in a free molecular flow
Kai Koike
2018, 11(3): 441-467 doi: 10.3934/krm.2018020 +[Abstract](32) +[HTML](20) +[PDF](498.76KB)

Motion of a rigid body immersed in a semi-infinite expanse of free molecular gas in a \begin{document}$ d$\end{document}-dimensional region bounded by an infinite plane wall is studied. The free molecular flow is described by the free Vlasov equation with the specular boundary condition. We show that the velocity \begin{document}$ V(t)$\end{document} of the body approaches its terminal velocity \begin{document}$ V_{∞}$\end{document} according to a power law \begin{document}$ V_{∞}-V(t)≈ t^{-(d-1)}$\end{document} by carefully analyzing the pre-collisions due to the presence of the wall. The exponent \begin{document}$ d-1$\end{document} is smaller than \begin{document}$ d+2$\end{document} for the case without the wall found in the classical work by Caprino, Marchioro and Pulvirenti [Comm. Math. Phys., 264 (2006), 167-189] and thus slower convergence results from the presence of the wall.

Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary
Linjie Xiong
2018, 11(3): 469-490 doi: 10.3934/krm.2018021 +[Abstract](33) +[HTML](26) +[PDF](497.17KB)

This paper concerns the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for isentropic fluids in a bounded domain with a Navier-slip boundary condition. In [2], it has been proved that if the velocity is imposed the homogeneous Dirichlet boundary condition, as the Mach number goes to 0, the velocity of the compressible flow converges strongly in \begin{document}$ L^2$\end{document} under the geometrical assumption (H) on the domain. We justify the same strong convergence when the slip length in the Navier condition is the reciprocal of the square root of the Mach number.

Kinetic description of collision avoidance in pedestrian crowds by sidestepping
Adriano Festa, Andrea Tosin and Marie-Therese Wolfram
2018, 11(3): 491-520 doi: 10.3934/krm.2018022 +[Abstract](31) +[HTML](22) +[PDF](3717.76KB)

In this paper we study a kinetic model for pedestrians, who are assumed to adapt their motion towards a desired direction while avoiding collisions with others by stepping aside. These minimal microscopic interaction rules lead to complex emergent macroscopic phenomena, such as velocity alignment in unidirectional flows and lane or stripe formation in bidirectional flows. We start by discussing collision avoidance mechanisms at the microscopic scale, then we study the corresponding Boltzmann-type kinetic description and its hydrodynamic mean-field approximation in the grazing collision limit. In the spatially homogeneous case we prove directional alignment under specific conditions on the sidestepping rules for both the collisional and the mean-field model. In the spatially inhomogeneous case we illustrate, by means of various numerical experiments, the rich dynamics that the proposed model is able to reproduce.

Landau damping in the multiscale Vlasov theory
Miroslav Grmela and Michal Pavelka
2018, 11(3): 521-545 doi: 10.3934/krm.2018023 +[Abstract](28) +[HTML](29) +[PDF](487.04KB)

Vlasov kinetic theory is extended by adopting an extra one particle distribution function as an additional state variable characterizing the micro-turbulence internal structure. The extended Vlasov equation keeps the reversibility, the Hamiltonian structure, and the entropy conservation of the original Vlasov equation. In the setting of the extended Vlasov theory we then argue that the Fokker-Planck type damping in the velocity dependence of the extra distribution function induces the Landau damping. The same type of extension is made also in the setting of fluid mechanics.

High order approximation for the Boltzmann equation without angular cutoff
Lingbing He and Yulong Zhou
2018, 11(3): 547-596 doi: 10.3934/krm.2018024 +[Abstract](32) +[HTML](22) +[PDF](603.19KB)

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules
Milana Pavić-Čolić and Maja Tasković
2018, 11(3): 597-613 doi: 10.3934/krm.2018025 +[Abstract](27) +[HTML](22) +[PDF](398.26KB)

We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its 1-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities
Tomasz Komorowski, Stefano Olla and Marielle Simon
2018, 11(3): 615-645 doi: 10.3934/krm.2018026 +[Abstract](27) +[HTML](33) +[PDF](519.62KB)

We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.

The rigorous derivation of the Linear Landau equation from a particle system in a weak-coupling limit
Nicolo' Catapano
2018, 11(3): 647-695 doi: 10.3934/krm.2018027 +[Abstract](31) +[HTML](35) +[PDF](1248.55KB)

We consider a system of N particles interacting via a short-range smooth potential, in a weak-coupling regime. This means that the number of particles \begin{document}$N$\end{document} goes to infinity and the range of the potential \begin{document}$ε$\end{document} goes to zero in such a way that \begin{document}$Nε^{2} = α$\end{document}, with \begin{document}$α$\end{document} diverging in a suitable way. We provide a rigorous derivation of the Linear Landau equation from this particle system. The strategy of the proof consists in showing the asymptotic equivalence between the one-particle marginal and the solution of the linear Boltzmann equation with vanishing mean free path. This point follows [3] and makes use of technicalities developed in [16]. Then, following the ideas of Landau, we prove the asympotic equivalence between the solutions of the Boltzmann and Landau linear equation in the grazing collision limit.

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