Global Weak Solutions of the Boltzmann Equation

A new definition of the concept of weak solution of the nonlinear Boltzmann equation is introduced. It is proved that, without any truncation on the collision kernel, the Boltzmann equation in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

Warm cascade states in a forced-dissipated Boltzmann gas of hard spheres

We study the homogeneous isotropic Boltzmann equation for an open system. For the case of a hard spheres gas, we look for nonequilibrium steady solutions in the presence of forcing and dissipation. Using the language of weak turbulence theory, we analyze the possibility of observing the Kolmogorov-Zakharov steady distributions, i.e. solutions characterized by constant fluxes of conserved quantities. We derive a differential approximation model and we find that the expected nonequilibrium steady solutions have always the form of warm cascades. We propose an analytical prediction for the relation between the forcing and dissipation and the thermodynamic quantities of the system. Specifically, we find that the temperature of the system is independent of the forcing amplitude and determined only by the forcing and dissipation scales. Finally, we perform direct numerical simulations of the Boltzmann equation finding consistent results with our theoretical predictions.     

The Limiting Kinetic Equation of the Consistent Boltzmann Algorithm for Dense Gases

This paper establishes a theoretical foundation for the Consistent Boltzmann Algorithm (CBA) by deriving the limiting kinetic equation. The formulation is similar to the proof by one of the authors that the Boltzmann equation is the limiting kinetic equation for Direct Simulation Monte Carlo [W. Wagner, J. Statist. Phys. 66:1011 (1992)]. For a simplified model distilled from CBA, the limiting equation is solved numerically, and very good agreement with the predictions of the theory is found.     

Weak Solutions of the Boltzmann Equation Without Angle Cutoff

The definition of the concept of weak solution of the nonlinear Boltzmann equation, recently introduced by the author, is used to prove that, without any cutoff in the collision kernel, the Boltzmann equation for Maxwell molecules in the one-dimensional case has a global weak solution in this sense. Global conservation of energy follows.     

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.     

On the Rate of Entropy Production for the Boltzmann Equation

We show that there exists a wide class of distribution functions (with moments of any order as close to their equilibrium values as we like) which can provide an abnormally low rate of entropy production. The result is valid for the Boltzmann equation with any cross section (|V|, ) satisfying a mild restriction. The functions are constructed in an explicit form and we discuss some applications of our results.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

Coarsely grained stochastic Boltzmann equation and its moment equations

The stochastic Boltzmann equation is coarsely grained. The coarsely grained stochastic (CGS) Boltzmann equation has fluctuating terms in its collision term. On the basis of the CGS Boltzmann equation, reduced Grad’s 26 moment equations are derived. Coarsely grained moment equations obtained from the CGS Boltzmann equation show that fluctuating terms remain as nonvanishing terms owing to the nonlinearity in the collision term of the CGS Boltzmann equation. The Navier-Stokes-Fourier law obtained using the CGS Boltzmann equation indicates that the pressure deviator and heat flux include fluctuations of their one-order higher moments.     

Exact Eternal Solutions of the Boltzmann Equation

We construct two families of self-similar solutions of the Boltzmann equation in an explicit form. They turn out to be eternal and positive. They do not possess finite energy. Asymptotic properties of the solutions are also studied.     

Reciprocal relations based on the non-stationary Boltzmann equation

The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.     

Factorization symmetry in the lattice Boltzmann method

A non-perturbative algebraic theory of the lattice Boltzmann method is developed based on the symmetry of a product. It involves three steps: (i) Derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher-order Gaussian moments, (ii) A special quasi-equilibrium distribution function found analytically in closed form on the product-lattice in two and three spatial dimensions, and which proves the factorization of quasi-equilibrium moments, and (iii) An algebraic method of pruning based on a one-into-one relation between groups of discrete velocities and moments. Two routes of constructing lattice Boltzmann equilibria are distinguished. The present theory includes previously known limiting and special cases of lattices, and enables automated derivation of lattice Boltzmann models from two-dimensional tables, by finding the roots of one polynomial and solving a few linear systems.     

Half-Space Problems for the Boltzmann Equation: A Survey

This paper reviews recent mathematical results on the half-space problem for the Boltzmann equation. The case of a phase transition is discussed in detail.     

Estimating the Solutions of the Boltzmann Equation

We discuss some possible estimates of the solutions of the Boltzmann equation, which might permit a progress in the theory of existence of weak solutions.     

A density-corrected quantum Boltzmann equation

Binary correlations are a recognized part of the pair density operator, but the influence of binary correlations on the singlet density operator is usually not emphasized. Here free motion and binary correlations are taken as independent building blocks for the structure of the nonequilibrium singlet and pair density operators. Binary correlations are assumed to arise from the collision of twofree particles. Together with the first BBGKY equation and a retention of all terms that are second order in gas density, a generalization of the Boltzmann equation is obtained. This is an equation for thefree particle density operator rather than for the (full) singlet density operator. The form for the pressure tensor calculated from this equation reduces at equilibrium to give the correct (Beth-Uhlenbeck) second virial coefficient, in contrast to a previous quantum Boltzmann equation, which gave only part of the quantum second virial coefficient. Generalizations to include higher-order correlations and collision types are indicated.     

On the asymptotic equivalence between the Enskog and the Boltzmann equations

An asymptotic equivalence theorem is proven between the solutions of the initial value problem in all space for the Boltzmann and Enskog equations for initial data which assure global existence for the solutions to the initial value problem for one of the two equations. The proof is given starting from the solution of the Boltzmann equation, then the proof line is simply indicated when one starts from the Enskog equation. The proof holds for Knudsen numbers of the order of unity and equivalence is proven when the scale of the dimensions of the gas particles characterizing the Enskog equation tends to zero.On leave from Department of Mathematics, University of Warsaw, Poland.     

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers' Equation

In this paper,we present a Cole-Hopf transformation based lattice Boltzmann(LB) model for solving one-dimensional Burgers' equation,and compared to available LB models,the effect of nonlinear convection term can be eliminated.Through Chapman-Enskog analysis,it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model.Some numerical tests are also performed to validate the present LB model,and the numerical results show that,similar to previous LB models,the present model also has a second-order convergence rate in space,but it is more accurate than the previous ones.     

Self-Similar Solutions of the Boltzmann Equation for Non-Maxwell Molecules

We show that the method previously used by the authors to obtain self-similar, eternal solutions of the space-homogeneous Boltzmann equation for Maxwell molecules yields different results when extended to other power-law potentials (including hard spheres). In particular, self-similar solutions cease to exist for a positive time for hard potentials. In the case of soft potentials, the solutions exist for all potive times, but are not eternal.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

The dynamics of coupled nonlinear model Boltzmann equations

A multispecies gas described by coupled nonlinear Boltzmann equations is studied as a dynamical system. Properties are determined of theN coupled nonlinear ODEs for the number densities obtained from the Boltzmann equations for the spatially uniform system ofN species undergoing binary scattering, removal, and regeneration in the presence of an external force field and a reservoir of background gas. The physically realizable setQ, the nonnegative cone in theN-dimensional phase space of species number densities, is established as invariant under the flow. The fixed-point equations for the ODEs are shown to be equivalent to 2 ^N linear systems, and conditions for the stability and instability of the fixed points are then established. Stable fixed points are demonstrated to exist inQ by showing that they enter via a sequence of transcritical bifurcations as physical parameters are varied. For the two-species case the typical global structure of the solutions is established. Various particular cases are described including one which possesses an infinite family of periodic solutions and one that depends delicately upon initial conditions due to a separatrix that separatesQ into two invariant sets.