Solutions of the Boltzmann equation for Maxwell interactions and singular angle-dependent cross sections

We consider the spatially homogeneous and isotropic Boltzmann distribution function in the case of nonisotropic, binary cross sections inversely proportional to the relative speed of the colliding particles. Further, we allow the angle dependence of the differential cross section() to be singular in the forward direction ( 0). We assume (), d < which includes the case of a Maxwellian interaction. We explicitly show how to construct the solutions of the Boltzmann equation, study their properties, and obtain for a class of solutions sufficient conditions for their existence at any positive time value. We extend the formalism to the more general case of arbitrary dimensionality. We observe an effect noticed previously by Krook, Wu, and Tjon in other models of the Boltzmann equations-namely, for special initial distributions, we find solutions which exhibit an excess of higher energy particles at later time.     

The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation

An existence and uniqueness result for the homogeneous Boltzmann hierarchy is proven, by exploiting the statistical solutions to the homogeneous Boltzmann equation.     

A steady state, supersonic flow solution of the Boltzmann equation

Nonclassical effects which appear in the supersonic nonequilibrium gas flow of a nonuniform relaxation described by the Boltzmann equation are studied. For such a flow, in particular, the heat flux and the temperature gradient have the same signs. Analytical and numerical results are presented. Possible experimental verification is discussed.     

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ^d , and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.     

Global solutions of the Boltzmann equation on a lattice

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.Supported by National Science Foundation Grant ENG-7515882.     

The Boltzmann equation for a polyatomic gas

A formulation of the kinetic theory of dilute, classical polyatomic gases is given which parallels the Waldmann development for structureless molecules. In the first section the Boltzmann equation is written in terms of the specific rates of inelastic collision processes and then the properties of these rates and those of the corresponding collision cross sections are examined. The dependence of the distribution function on the dynamical variables is discussed and the equations of change for the gas are derived. Finally, a study is made of the properties of the linearized Boltzmann collision operation. In the second section the Boltzmann equation is deduced from a rigorous statistical-mechanical point of view and discussed in terms of the basic ideas of Bogoliubov. The computationally important special case of impulsive interactions is then considered.This research was supported in part by a grant from the National Science Foundation and in part by the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution No. 2554.     

Stationary Boltzmann equation for a degenerate gas in a slab: Boundary value problem and hydrodynamics

The boundary value problem for the stationary Boltzmann equation for a model gas in a plane slab is solved in full generality. The asymptotic behavior as the size of the slab goes to infinity is studied via a Chapman-Enskog expansion.     

Iterative solution of the Boltzmann equation

We define an iterative scheme to solve the nonlinear Boltzmann equation. Conservation rules are maintained at each iterative step. We apply this method to a spatially uniform and isotropic velocity distribution function on the Maxwell and very-hard-particle models. A particular example is evaluated and results are compared with the exact solution. It shows to be a very fast convergent approach.     

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.     

Exact solutions of the Boltzmann equation in the VHP model with removal interaction

The linear and nonlinear Boltzmann equation for very hard particles (VHP) is considered in the case when the collision between two particles may lead not only to elastic scattering, but also to a removal event with the disappearance of the molecules. The extended transport equation is solved for arbitrary initial distributions. The computations are carried out explicitly for a special class of initial distributions and for various removal rates. The results are demonstrated graphically. Finally, source terms fulfilling physically reasonable conditions are introduced into the VHP model, and the time-dependent particle number is calculated.     

Boltzmann equation for a dissociating gas

The incorporation of three-body collisions for dissociation/recombination into the Boltzmann equation is discussed. Conditions are assumed such that collisions are completed in the sense of scattering theory, so the collision operator is determined by scattering and reaction cross sections. The resulting equation has anH-theorem, and the equilibrium solution requires the law of mass action in addition to the Maxwellian dependence on momentum. A brief discussion is given of the normal solution and the transport coefficients.This paper is dedicated to Prof. E. G. D. Cohen on the occasion of his 65th birthday.     

On the Boltzmann equation for the Lorentz gas

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter >0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.     

Global existence of solutions for a model Boltzmann equation

A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finiteH-functional and belonging toL ¹ (L _x ). Methods previously introduced by Tartar to deal with discrete velocity models are used.     

Global existence in L1 for the Enskog equation and convergence of the solutions to solutions of the Boltzmann equation

For the Enskog equation in a box an existence theorem is proved for initial data with finite mass, energy, and entropy. Then, by letting the diameter of the molecules go to zero, the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation is proved.     

Boltzmann Equations For Mixtures of Maxwell Gases: Exact Solutions and Power Like Tails

We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.     

Boltzmann equation on a lattice: Existence and uniqueness of solutions

Cercignani, Greenberg, and Zweifel proved the existence and uniqueness of solutions of the Boltzmann equation on a toroidal lattice under the assumption that the collision kernel is bounded. We give an alternative, considerably simpler, proof which is based on a fixed point argument.     

Nonisotropic solutions of the Boltzmann equation

We consider the relaxation to equilibrium of a spatially uniform Maxwellian gas. We expand the solution of the nonlinear Boltzmann equation in a truncated series of orthogonal functions. We integrate numerically the equation for non-isotropic initial conditions. For certain simple conditions we find interesting proximity effects and other transient relaxation phenomena at thermal energies. Furthermore, we define a resummation of the orthogonal expansion which is more convenient than the original one for the numerical analysis of the relaxation process.     

The hilbert expansion to the Boltzmann equation for steady flow

The Hilbert expansion to the Boltzmann equation is carried out for steady flow. It is shown that the first term in the Hubert series for the distribution function is a local Maxwellian leading to the steady Euler equations. The steady field equations that follow from the solution of the second term in the series are derived. The formulas for thermal conductivity and for viscosity of Hilbert that appear in the steady field equations of the second approximation are shown to be precisely the same as those obtained by Chapman and Enskog. The procedure to obtain higher approximations by Hubert's method is summarized.     

Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, periodic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L¹-norm for any initial data with the finite total mass, mechanical energy and entropy.