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Kinetic & Related Models

February 2018 , Volume 11 , Issue 1

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Hypocoercive estimates on foliations and velocity spherical Brownian motion
Fabrice Baudoin and Camille Tardif
2018, 11(1): 1-23 doi: 10.3934/krm.2018001 +[Abstract](1066) +[HTML](149) +[PDF](417.3KB)
Abstract:

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in H1 and in L2.

A non-relativistic model of plasma physics containing a radiation reaction term
Sebastian Bauer
2018, 11(1): 25-42 doi: 10.3934/krm.2018002 +[Abstract](1050) +[HTML](106) +[PDF](458.8KB)
Abstract:

While a fully relativistic collisionless plasma is modeled by the Vlasov-Maxwell system a good approximation in the non-relativistic limit is given by the Vlasov-Poisson system. We modify the Vlasov-Poisson system so that damping due to the relativistic effect of radiation reaction is included. We prove existence and uniqueness as well as higher regularity of local classical solutions. Our results also include the higher regularity of classical solutions of the Vlasov-Poisson system depending on the regularity of the initial datum.

A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures
Etienne Bernard, Laurent Desvillettes, Franç cois Golse and Valeria Ricci
2018, 11(1): 43-69 doi: 10.3934/krm.2018003 +[Abstract](1005) +[HTML](101) +[PDF](449.0KB)
Abstract:

In this paper, we formally derive the thin spray equation for a steady Stokes gas (i.e. the equation consists in a coupling between a kinetic — Vlasov type — equation for the dispersed phase and a — steady — Stokes equation for the gas). Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard, Desvillettes, Golse, Ricci, Commun.Math.Sci.,15 (2017), 1703-1741] wherethe evolution of the gas is governed by the Navier-Stokes equation.

Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics
Marzia Bisi, Tommaso Ruggeri and Giampiero Spiga
2018, 11(1): 71-95 doi: 10.3934/krm.2018004 +[Abstract](1068) +[HTML](130) +[PDF](527.8KB)
Abstract:

The aim of this paper is to compare different kinetic approaches to a polyatomic rarefied gas: the kinetic approach via a continuous energy parameter $I$ and the mixture-like one, based on discrete internal energy. We prove that if we consider only $6$ moments for a non-polytropic gas the two approaches give the same symmetric hyperbolic differential system previously obtained by the phenomenological Extended Thermodynamics. Both meaning and role of dynamical pressure become more clear in the present analysis.

Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum
Jishan Fan and Yueling Jia
2018, 11(1): 97-106 doi: 10.3934/krm.2018005 +[Abstract](1099) +[HTML](142) +[PDF](324.7KB)
Abstract:

In this paper, we prove the local well-posedness of strong solutions for a compressible Navier-Stokes-Maxwell system, provided the initial data satisfy a natural compatibility condition. We do not assume the positivity of initial density, it may vanish in an open subset (vacuum) of $Ω$.

Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics
Moon-Jin Kang
2018, 11(1): 107-118 doi: 10.3934/krm.2018006 +[Abstract](952) +[HTML](93) +[PDF](327.9KB)
Abstract:

We investigate a non-contraction property of large perturbations around intermediate entropic shock waves and contact discontinuities for the three-dimensional planar compressible isentropic magnetohydrodynamics (MHD). To do that, we take advantage of criteria developed by the author and Vasseur in [6], and non-contraction property is measured by pseudo distance based on relative entropy.

Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
Victor A. Kovtunenko and Anna V. Zubkova
2018, 11(1): 119-135 doi: 10.3934/krm.2018007 +[Abstract](1272) +[HTML](225) +[PDF](452.3KB)
Abstract:

In this paper a mathematical model generalizing Poisson-Nernst-Planck system is considered. The generalized model presents electrokinetics of species in a two-phase medium consisted of solid particles and a pore space. The governing relations describe cross-diffusion of the charged species together with the overall electrostatic potential. At the interface between the pore and the solid phases nonlinear electro-chemical reactions are taken into account provided by jumps of field variables. The main advantage of the generalized model is that the total mass balance is kept within our setting. As the result of the variational approach, well-posedness properties of a discontinuous solution of the problem are demonstrated and supported by the energy and entropy estimates.

The derivation of the linear Boltzmann equation from a Rayleigh gas particle model
Karsten Matthies, George Stone and Florian Theil
2018, 11(1): 137-177 doi: 10.3934/krm.2018008 +[Abstract](1025) +[HTML](105) +[PDF](427.3KB)
Abstract:

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.

Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
Shuguang Shao, Shu Wang and Wen-Qing Xu
2018, 11(1): 179-190 doi: 10.3934/krm.2018009 +[Abstract](1027) +[HTML](108) +[PDF](376.0KB)
Abstract:

In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of \begin{document} ${\partial _t}u +(\mathcal {D}^{-1/2}u)·\nabla u + \nabla p =-\mathcal {A}^2u$ \end{document}, where \begin{document} $\mathcal {D}$ \end{document} and \begin{document} $\mathcal {A}$ \end{document} are Fourier multipliers defined by \begin{document} $\mathcal {D}=|\nabla|$ \end{document} and \begin{document} $\mathcal {A}= |\nabla|\ln^{-1/4}(e + λ \ln (e + |\nabla|)) $ \end{document} with \begin{document} $λ≥q0$ \end{document}. The symbols of the \begin{document} $\mathcal {D}$ \end{document} and \begin{document} $\mathcal {A}$ \end{document} are \begin{document} $m(ξ) =\left| ξ \right|$ \end{document} and \begin{document} $h(ξ) = \left| ξ \right| / g(ξ)$ \end{document} respectively, where \begin{document} $g(ξ) = {\ln ^{{1 / 4}}}(e + λ \ln (e + |ξ|))$ \end{document}, \begin{document} $λ≥0$ \end{document}. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that \begin{document} $h(ξ) =|ξ|^α$ \end{document} for \begin{document} $α≥q 5/4$ \end{document}. Here by changing the advection term we greatly weaken the dissipation to \begin{document} $ h(ξ)={{\left| ξ \right|} / g(ξ)}$ \end{document}. We prove the global well-posedness for any smooth initial data in \begin{document} $H^s(\mathbb{R}^3)$ \end{document}, \begin{document} $ s≥q3 $ \end{document} by using the energy method.

Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$
Zhong Tan and Leilei Tong
2018, 11(1): 191-213 doi: 10.3934/krm.2018010 +[Abstract](1180) +[HTML](99) +[PDF](477.2KB)
Abstract:

The compressible non-isentropic Navier-Stokes-Maxwell system is investigated in \begin{document} $\mathbb{R}^3$ \end{document} and the global existence and large time behavior of solutions are established by pure energy method provided the initial perturbation around a constant state is small enough. We first construct the global unique solution under the assumption that the \begin{document} $H^3$ \end{document} norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to \begin{document} $\dot{H}^{-s}$ \end{document} (\begin{document} $0≤ s<3/2$ \end{document}) or \begin{document} $\dot{B}_{2, ∞}^{-s}$ \end{document} (\begin{document} $0< s≤3/2$ \end{document}), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the \begin{document} $L^p$ \end{document}-\begin{document} $L^2$ \end{document} \begin{document} $(1≤ p≤ 2)$ \end{document} type of the decay rates follows without requiring that the \begin{document} $L^p$ \end{document} norm of initial data is small.

Letter to the editors in chief
Tai-Ping Liu and Shih-Hsien Yu
2018, 11(1): 215-217 doi: 10.3934/krm.2018011 +[Abstract](1756) +[HTML](261) +[PDF](273.0KB)
Abstract:

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