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Kinetic & Related Models

2016 , Volume 9 , Issue 3

Issue in memory of Alfredo Lorenzi

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A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions
Yingshan Chen and Mei Zhang
2016, 9(3): 429-441 doi: 10.3934/krm.2016001 +[Abstract](387) +[PDF](422.0KB)
Abstract:
In this paper, we establish a new blowup criterion for the strong solutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In [13], Wen, Yao, and Zhu prove that if the strong solutions blow up at finite time $T^*$, the mass in $L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extend Wen, Yao, and Zhu's work in the sense of the concentration of mass in $BMO(\Omega)$ norm at $T^*$. The method can be applied to study the blow-up criterion in terms of the concentration of density in $BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in [11]. Moreover, the appearance of vacuum is allowed in the paper.
Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain
Jishan Fan, Fucai Li and Gen Nakamura
2016, 9(3): 443-453 doi: 10.3934/krm.2016002 +[Abstract](604) +[PDF](337.8KB)
Abstract:
In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.
Separated characteristics and global solvability for the one and one-half dimensional Vlasov Maxwell system
Robert Glassey, Stephen Pankavich and Jack Schaeffer
2016, 9(3): 455-467 doi: 10.3934/krm.2016003 +[Abstract](357) +[PDF](348.7KB)
Abstract:
The motion of a collisionless plasma - a high-temperature, low-density, ionized gas - is described by the Vlasov-Maxwell (VM) system. These equations are considered in one space dimension and two momentum dimensions without the assumption of relativistic velocity corrections. The main results are bounds on the spatial and velocity supports of the particle distribution function and uniform estimates on derivatives of this function away from the critical velocity $\vert v_1 \vert = 1$. Additionally, for initial particle distributions that are even in the second velocity argument $v_2$, the global-in-time existence of solutions is shown.
Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients
Bingkang Huang, Lusheng Wang and Qinghua Xiao
2016, 9(3): 469-514 doi: 10.3934/krm.2016004 +[Abstract](383) +[PDF](628.0KB)
Abstract:
We study the nonlinear stability of rarefaction waves to the Cauchy problem of one-dimensional compressible Navier-Stokes equations for a viscous and heat conducting ideal polytropic gas when the transport coefficients depend on both temperature and density. When the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially, we show that rarefaction waves are nonlinear stable provided that $(\gamma- 1)\cdot H^3(\mathbb{R})$-norm of the initial perturbation is suitably small with $\gamma>1$ being the adiabatic gas constant.
The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction
Shuangqian Liu and Qinghua Xiao
2016, 9(3): 515-550 doi: 10.3934/krm.2016005 +[Abstract](339) +[PDF](572.9KB)
Abstract:
We are concerned with the Cauchy problem of the relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. For perturbative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov-Maxwell-Boltzmann system, and also obtain as a byproduct the convergence rates of solutions. Our proof is based on a new time-velocity weighted energy method and some optimal temporal decay estimates on the solution itself. The results also extend the case of ``hard ball" model considered by Guo and Strain [Comm. Math. Phys. 310: 49--673 (2012)] to the short range interactions.
Local well-posedness for the tropical climate model with fractional velocity diffusion
Caochuan Ma, Zaihong Jiang and Renhui Wan
2016, 9(3): 551-570 doi: 10.3934/krm.2016006 +[Abstract](473) +[PDF](443.2KB)
Abstract:
This paper deals with the Cauchy problem for tropical climate model with the fractional velocity diffusion which was derived by Frierson-Majda-Pauluis in [16]. We establish the local well-posedness of strong solutions to this generalized model.
A kinetic reaction model: Decay to equilibrium and macroscopic limit
Lukas Neumann and Christian Schmeiser
2016, 9(3): 571-585 doi: 10.3934/krm.2016007 +[Abstract](368) +[PDF](401.1KB)
Abstract:
We propose a kinetic relaxation-model to describe a generation-recombination reaction of two species. The decay to equilibrium is studied by two recent methods [9,13] for proving hypocoercivity of the linearized equations. Exponential decay of small perturbations can be shown for the full nonlinear problem. The macroscopic/fast-reaction limit is derived rigorously employing entropy decay, resulting in a nonlinear diffusion equation for the difference of the position densities.
Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma
Masahiro Suzuki
2016, 9(3): 587-603 doi: 10.3934/krm.2016008 +[Abstract](389) +[PDF](451.7KB)
Abstract:
The main concern of this paper is to analyze a boundary layer called a sheath that occurs on the surface of materials when in contact with a multicomponent plasma. For the formation of a sheath, the generalized Bohm criterion demands that ions enter the sheath region with a high velocity. The motion of a multicomponent plasma is governed by the Euler--Poisson equations, and a sheath is understood as a monotone stationary solution to those equations. In this paper, we prove the unique existence of the monotone stationary solution by assuming the generalized Bohm criterion. Moreover, it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in weighted Sobolev space. We also obtain the convergence rate, which is subject to the decay rate of the initial perturbation, of the time global solution toward the stationary solution.
Entropy production for ellipsoidal BGK model of the Boltzmann equation
Seok-Bae Yun
2016, 9(3): 605-619 doi: 10.3934/krm.2016009 +[Abstract](367) +[PDF](412.9KB)
Abstract:
The ellipsoidal BGK model (ES-BGK) is a generalized version of the original BGK model, designed to yield the correct Prandtl number in the Navier-Stokes limit. In this paper, we make two observations on the entropy production functional of the ES-BGK model. First, we show that the Cercignani type estimate holds for the ES-BGK model in the whole range of relaxation parameter $-1/2<\nu<1$. Secondly, we observe that the ellipsoidal relaxation operator satisfies an unexpected sign-definite property. Some implications of these observations are also discussed.

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