# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

February 2018 , Volume 11 , Issue 1

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2018, 11(1): 1-23 doi: 10.3934/krm.2018001 +[Abstract](2293) +[HTML](168) +[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.

2018, 11(1): 25-42 doi: 10.3934/krm.2018002 +[Abstract](2158) +[HTML](119) +[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.

2018, 11(1): 43-69 doi: 10.3934/krm.2018003 +[Abstract](2469) +[HTML](118) +[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.

2018, 11(1): 71-95 doi: 10.3934/krm.2018004 +[Abstract](2262) +[HTML](145) +[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.

2018, 11(1): 97-106 doi: 10.3934/krm.2018005 +[Abstract](2307) +[HTML](155) +[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 $Ω$.

2018, 11(1): 107-118 doi: 10.3934/krm.2018006 +[Abstract](1866) +[HTML](107) +[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.

2018, 11(1): 119-135 doi: 10.3934/krm.2018007 +[Abstract](2613) +[HTML](620) +[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.

2018, 11(1): 137-177 doi: 10.3934/krm.2018008 +[Abstract](2401) +[HTML](119) +[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.

2018, 11(1): 179-190 doi: 10.3934/krm.2018009 +[Abstract](2311) +[HTML](123) +[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.

2018, 11(1): 191-213 doi: 10.3934/krm.2018010 +[Abstract](2700) +[HTML](118) +[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.

2018, 11(1): 215-217 doi: 10.3934/krm.2018011 +[Abstract](3025) +[HTML](299) +[PDF](273.0KB)
Abstract:

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