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Kinetic & Related Models

April 2020 , Volume 13 , Issue 2

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Stochastic Cucker-Smale flocking dynamics of jump-type
Martin Friesen and Oleksandr Kutoviy
2020, 13(2): 211-247 doi: 10.3934/krm.2020008 +[Abstract](1126) +[HTML](135) +[PDF](533.06KB)
Abstract:

We present a stochastic version of the Cucker-Smale flocking dynamics described by a system of \begin{document}$ N $\end{document} interacting particles. The velocity aligment of particles is purely discontinuous with unbounded and, in general, non-Lipschitz continuous interaction rates. Performing the mean-field limit as \begin{document}$ N \to \infty $\end{document} we identify the limiting process with a solution to a nonlinear martingale problem associated with a McKean-Vlasov stochastic equation with jumps. Moreover, we show uniqueness and stability for the kinetic equation by estimating its solutions in the total variation and Wasserstein distance. Finally, we prove uniqueness in law for the McKean-Vlasov equation, i.e. we establish propagation of chaos.

On Fokker-Planck equations with In- and Outflow of Mass
Martin Burger, Ina Humpert and Jan-Frederik Pietschmann
2020, 13(2): 249-277 doi: 10.3934/krm.2020009 +[Abstract](972) +[HTML](138) +[PDF](1670.08KB)
Abstract:

Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.

The BGK approximation of kinetic models for traffic
Michael Herty, Gabriella Puppo, Sebastiano Roncoroni and Giuseppe Visconti
2020, 13(2): 279-307 doi: 10.3934/krm.2020010 +[Abstract](900) +[HTML](141) +[PDF](5653.39KB)
Abstract:

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.

Trend to the equilibrium for the Fokker-Planck system with an external magnetic field
Zeinab Karaki
2020, 13(2): 309-344 doi: 10.3934/krm.2020011 +[Abstract](1219) +[HTML](137) +[PDF](619.95KB)
Abstract:

We consider the Fokker-Planck equation with an external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like certain Lebesgue spaces with polynomial weights and modified weighted Sobolev spaces, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

Diffusion and kinetic transport with very weak confinement
Emeric Bouin, Jean Dolbeault and Christian Schmeiser
2020, 13(2): 345-371 doi: 10.3934/krm.2020012 +[Abstract](1142) +[HTML](160) +[PDF](435.49KB)
Abstract:

This paper is devoted to Fokker-Planck and linear kinetic equations with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diffusion wins over the confinement although the potential diverges at infinity. When there is no confinement potential, it is possible to rely on Fourier analysis and mode-by-mode estimates for the kinetic equations. Here we develop an alternative approach based on moment estimates and Caffarelli-Kohn-Nirenberg inequalities of Nash type for diffusion and kinetic equations.

Global existence and long time behavior of the Ellipsoidal-Statistical-Fokker-Planck model for diatomic gases
Lei Jing and Jiawei Sun
2020, 13(2): 373-400 doi: 10.3934/krm.2020013 +[Abstract](771) +[HTML](134) +[PDF](414.61KB)
Abstract:

We are concerned with the global existence and long time behavior of the solutions to the ES-FP model for diatomic gases proposed in [22]. The global existence of the solutions for this model near the global Maxwellian is established by nonlinear energy method based on the macro-micro decomposition. An algebraic convergence rate in time of the solutions to the equilibrium state is obtained by constructing the compensating function. Since the density distribution function \begin{document}$ F(t, x, v, I) $\end{document} also contains energy variable \begin{document}$ I $\end{document}, we derive more general Poincaré inequality including variables \begin{document}$ v, I $\end{document} on \begin{document}$ \mathbb{R}^3\times \mathbb{R}^+ $\end{document} to establish the coercivity estimate of the linearized operator.

Asymptotic behavior of a second-order swarm sphere model and its kinetic limit
Dohyun Kim
2020, 13(2): 401-434 doi: 10.3934/krm.2020014 +[Abstract](1166) +[HTML](149) +[PDF](482.03KB)
Abstract:

We study the asymptotic behavior of a second-order swarm model on the unit sphere in both particle and kinetic regimes for the identical cases. For the emergent behaviors of the particle model, we show that a solution to the particle system with identical oscillators always converge to the equilibrium by employing the gradient-like flow approach. Moreover, we establish the uniform-in-time \begin{document}$ \ell_2 $\end{document}-stability using the complete aggregation estimate. By applying such uniform stability result, we can perform a rigorous mean-field limit, which is valid for all time, to derive the Vlasov-type kinetic equation on the phase space. For the proposed kinetic equation, we present the global existence of measure-valued solutions and emergent behaviors.

2019  Impact Factor: 1.311

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