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Kinetic and Related Models

June 2022 , Volume 15 , Issue 3

Special Issue on nonlinear waves and kinetic theory dedicated to the memory of Bob Glassey. Part I

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Yan Guo, Toan T. Nguyen and Walter A. Strauss
2022, 15(3): ⅰ-ⅱ doi: 10.3934/krm.2022011 +[Abstract](305) +[HTML](101) +[PDF](338.38KB)
The inviscid limit for the 2D Navier-Stokes equations in bounded domains
Claude W. Bardos, Trinh T. Nguyen, Toan T. Nguyen and Edriss S. Titi
2022, 15(3): 317-340 doi: 10.3934/krm.2022004 +[Abstract](534) +[HTML](163) +[PDF](562.37KB)

We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary.

A toy model for the relativistic Vlasov-Maxwell system
Jonathan Ben-Artzi, Stephen Pankavich and Junyong Zhang
2022, 15(3): 341-354 doi: 10.3934/krm.2021053 +[Abstract](409) +[HTML](134) +[PDF](374.7KB)

The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.

Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system
José A. Carrillo, Young-Pil Choi and Yingping Peng
2022, 15(3): 355-384 doi: 10.3934/krm.2021052 +[Abstract](533) +[HTML](123) +[PDF](564.48KB)

We provide a quantitative asymptotic analysis for the nonlinear Vlasov–Poisson–Fokker–Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov–Poisson–Fokker–Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.

Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space
Yunbai Cao and Chanwoo Kim
2022, 15(3): 385-401 doi: 10.3934/krm.2021034 +[Abstract](488) +[HTML](211) +[PDF](397.18KB)

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.

Phase mixing for solutions to 1D transport equation in a confining potential
Sanchit Chaturvedi and Jonathan Luk
2022, 15(3): 403-416 doi: 10.3934/krm.2022002 +[Abstract](366) +[HTML](125) +[PDF](494.58KB)

Consider the linear transport equation in 1D under an external confining potential \begin{document}$ \Phi $\end{document}:

For \begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} (with \begin{document}$ \varepsilon >0 $\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}, with an inverse polynomial decay rate \begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $\end{document}D under the external potential \begin{document}$ \Phi $\end{document}.

From kinetic to fluid models of liquid crystals by the moment method
Pierre Degond, Amic Frouvelle and Jian-Guo Liu
2022, 15(3): 417-465 doi: 10.3934/krm.2021047 +[Abstract](1053) +[HTML](150) +[PDF](894.7KB)

This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.

Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition
Hongjie Dong, Yan Guo and Timur Yastrzhembskiy
2022, 15(3): 467-516 doi: 10.3934/krm.2022003 +[Abstract](491) +[HTML](143) +[PDF](798.66KB)

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.

The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles
François Golse
2022, 15(3): 517-534 doi: 10.3934/krm.2022001 +[Abstract](400) +[HTML](117) +[PDF](449.81KB)

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.

2021 Impact Factor: 1.398
5 Year Impact Factor: 1.685
2021 CiteScore: 2.7




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