Kinetic and Related Models最新文献

We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed begin{document}$ {{mathfrak{c}}}>0 $end{document}. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than begin{document}$ {{mathfrak{c}}} $end{document}, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed begin{document}$ {{{{mathfrak{c}}}^ast}}>0 $end{document} such that if begin{document}$ {{mathfrak{c}}}geq{{{{mathfrak{c}}}^ast}} $end{document}, the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of begin{document}$ {{{{mathfrak{c}}}^ast}} $end{document} is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.

The free transport operator of probability density function begin{document}$ f(t, x, v) $end{document} is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high order regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condition. We obtain initial-boundary compatibility conditions for begin{document}$ C^{1}_{t, x, v} $end{document} and begin{document}$ C^{2}_{t, x, v} $end{document} regularity of the solution. We also provide regularity estimates.

In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48-55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit.

We establish existence of finite energy weak solutions to the kinetic Fokker-Planck equation and the linear Landau equation near Maxwellian, in the presence of specular reflection boundary condition for general domains. Moreover, by using a method of reflection and the begin{document}$ S_p $end{document} estimate of [7], we prove regularity in the kinetic Sobolev spaces begin{document}$ S_p $end{document} and anisotropic Hölder spaces for such weak solutions. Such begin{document}$ S_p $end{document} regularity leads to the uniqueness of weak solutions.

By employing the Fourier transform to derive key a priori estimates for the temporal gradient of the chemical signal, we establish the existence of global solutions and hydrodynamic limit of a chemotactic kinetic model with internal states and temporal gradient in one dimension, which is a system of two transport equations coupled to a parabolic equation proposed in [4].

Following closely the classical works [5]-[7] by Glassey, Strauss, and Schaeffer, we present a version of the Glassey-Strauss representation for the Vlasov-Maxwell systems in a 3D half space when the boundary is the perfect conductor.