With the existence and uniqueness of a vector value solution for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions proved [8], we present in this manuscript several properties for such a solution. We start by proving the gain of integrability of the gain term of the multispecies collision operator, extending the work done previously for the single species case [3]. In addition, we study the integrability properties of the multispecies collision operator as a bilinear form, revisiting and expanding the work done for a single gas [2]. With these estimates, together with a control by below for the loss term of the collision operator as in [8], we develop the propagation for the polynomially and exponentially $β$-weighted $L^p_β$-norms for the vector value solution. Finally, we extend such $L^p_β$-norms propagation property to $p=\infty$.
This document presents a priori estimates related to statistical moments and integrability properties for solutions of systems of monatomic gas mixtures modelled with the homogeneous Boltzmann equation with long range interactions for hard potentials. We detail the conditions for the generation and propagation of polynomial and exponential moments, and the integrability in Lebesgue spaces.
Multi-species Boltzmann equations for gaseous mixtures, with analytic cross sections and under Grad's angular cutoff assumption, are considered under diffusive scaling. In the limit, we formally obtain an explicit expression for the binary diffusion coefficients in the Maxwell-Stefan equations.
The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the operator for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable, is studied. With a probabilistic formulation of the collision operator as a starting point, compactnes…
We study a kinetic model for non-reactive mixtures of monatomic gases with hard-sphere cross-sections under isothermal condition. By considering a diffusive scaling of the kinetic model and using the method of moments, we formally obtain from the continuity and momentum balance equations of the species, in the limit as the scaling parameter goes to zero, the Maxwell-Stefan diffusion equations, with an explicit expression for the diffusion coefficients.
We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{ρ_{0}(t,x)}{(2πθ_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2θ_{0}(t,x)},\text{\ }θ_{0}(t,x)=Kρ_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $ρ_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson …
A mathematical model is proposed where the classical Maxwell-Stefan diffusion model for gas mixtures is coupled to an advection-type equation for the temperature of the physical system. This coupled system is derived from first principles in the sense that the starting point of our analysis is a system of Boltzmann equations for gaseous mixtures. We perform an asymptotic analysis on the Boltzmann model under diffuse scaling to arrive at the proposed coupled system.
In this paper, we rigorously derive a Boltzmann equation for mixtures from the many body dynamics of two types of hard sphere gases. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases. As a corollary of the derivation, we prove Boltzmann's propagation of chaos assumption for the case of a mixtures of gases.
We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.
Gyeongsang National University（庆尚国立大学）；Sangmyung University（祥明大学）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）；Sungkyunkwan University（成均馆大学）
Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has …
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the $3$-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in $L^1_vL^\infty_x(m)$, where $m\sim (1+|v|^k)$ is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for…
Global Well-posedness for the Multi-species Boltzmann Equation with Large Amplitude Initial Data
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）；Center for Mathematical Machine Learning and its Applications (CM2LA), Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学学院数学机器学习及其应用中心）
We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and $H$-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficinets of the local equilibiriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under …
The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in [42]. It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-tr…
We consider the socalled Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of mixture a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom li…
We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator $\mathbf{L}$ is known to possess an explicit spectral gap $λ_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. We study a new operator $\mathbf{L^\varepsilon}$ obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $\varep…
In this short 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, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation.
We investigate the diffusion asymptotics of the Boltzmann equation for gaseous mixtures, in the perturbative regime around a local Maxwellian vector whose fluid quantities solve a flux-incompressible Maxwell-Stefan system. Our framework is the torus and we consider hard-potential collision kernels with angular cutoff. As opposed to existing results about hydrodynamic limits in the mono-species case, the local Maxwellian we study here is not a local equilibrium of the mixture due to cross-interactions. By means of a hypocoercive formalism and introducing a suitable modified Sobolev norm, we bui…
We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green's function of the Boltzmann equation for the single species hard sphere collision model as developed in work of Liu and Yu. The cross species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.
We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.
We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally pro…
We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in $L^1$ obtained by the so-called "entropy-entropy production" method in the context of dissipative systems with slowly growing a priori bounds (see reference [14]).
In this paper, we present a splitting algorithm to solve multicomponent transport models. These models are related to plasma simulations, in which we consider the local thermodynamic equilibrium and weakly ionised plasma-mixture models that are used for medical and technical applications in etching processes. These multi-component transport-mixture models can be derived by approximately solving a linearised multi-component Boltzmann equation with an approximation of the collision terms in the mass, momentum and energy equations. The model-equations are nonlinear partial differential equations …
