Initial-Boundary value problems for the Boltzmann equation: Global existence of weak solutions

For an open set of ³ bounded or not, we consider initial-boundary value problems for the Boltzmann equation. For general gas-surface interaction laws and for hard potentials, we prove a global existence result for weak solutions. The proof uses the regularization of the collision operator and the renormalization method for the regularized problem. By using weak compactness in L¹ and averaged stability ofQ(f,f), we prove the existence of weak solutions of our problem.Dedicated to the Memory of Ronald DiPerna     

Model Boltzmann equation for gas mixtures: Construction and numerical comparison

A new model for the nonlinear Boltzmann equation for gas mixtures is constructed by the method employed in the derivation of the McCormack model in the linearized kinetic theory [F.J. McCormack, Phys. Fluids 16 (1973) 2095]. Then it is compared numerically with other existing models proposed in [P. Andries, K. Aoki, B. Perthame, J. Stat. Phys. 106 (2002) 993] and in [L.H. Holway Jr., Phys. Fluids 9 (1966) 1658] (the so-called ES-BGK model) as well as with the original Boltzmann equation. The new model is not restricted to the Maxwell molecule, can fit to general molecular models, and reproduces well solutions of the Boltzmann equation at least in the case of weak nonequilibrium. The numerical comparison is performed in the case of a binary gas mixture consisted of the hard-sphere or pseudo Maxwell molecules, after parameters concerning the molecular interaction are adjusted appropriately.     

Construction of solutions of the Boltzmann kinetic equation in a Knudsen layer

We consider the moment equation method for solving the Boltzmann equation in a Knudsen layer; the calculation of one of the moments of the collision integral is presented.     

Global weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

It is proved that under a physically realistic truncation of the collision kernel, the Boltzmann equation in the one-dimensional slab [0,1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. This solution satisfies the boundary conditions almost everywhere, and has, at worst, exponentially growing total energy.     

Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Based on the Bhatnagar–Gross–Krook (BGK) Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non‐linear source terms. Based on the unsteady time‐splitting technique and the non‐oscillatory, containing no free parameters, and dissipative (NND) finite‐difference method, the gas kinetic finite‐difference second‐order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm for the gas dynamical problems from various flow regimes is developed. To test the reliability of the present numerical method, the one‐dimensional shock‐tube problems and the flows past two‐dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related flows indicate that both high resolution of the flow fields and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Exact solution of the Boltzmann equation

We build up immediate connection between the nonlinear Boltzmann transport equation and the linear AKNS equation, and classify the Boltzmann equation as the Dirac equation by a new method for solving the Boltzmann equation out of keeping with the Chapman, Enskog and Grad’s way in this paper. Without the effect of other external fields, the exact solution of the Boltzmann equation can be obtained by the inverse scattering method.     

Numerical solutions of coupled Burgers' equation

Two new algorithms based on cubic spline function technique are proposed for solving Burgers' equation in one space variable and coupled Burgers' equation in two space variables. The algorithms have been analysed for their stability and convergence. Two test examples have been solved for illustrating the merits of the proposed numerical method. The method can be extended for solving non-linear problems arising in mechanics and other areas.     

Numerical computation of solitary wave solutions of the Rosenau equation

We construct numerically solitary wave solutions of the Rosenau equation using the Petviashvili iteration method. We first summarize the theoretical results available in the literature for the existence of solitary wave solutions. We then apply two numerical algorithms based on the Petviashvili method for solving the Rosenau equation with single or double power law nonlinearity. Numerical calculations rely on a uniform discretization of a finite computational domain. Through some numerical experiments we observe that the algorithm converges rapidly and it is robust to very general forms of the initial guess.     

Numerical solutions for singularly perturbed semi-linear parabolic equation

In this paper,we discuss singularly perturbed semi-linear parabolic equations for one dimension and two dimension,we find numerical solutions by using both the line-method and the exact difference scheme on a special non-uniform discretization mesh.The uniform convergence in e of the first order accuracy is obtained.     

Conditions of applicability of the Boltzmann equation

The existence of certain characteristic times, introduced by Bogolyubov [1], is of fundamental importance for the derivation of the Boltzmann equation from the Liouville equations. In the present paper characteristic spatial scales are also introduced, which permit a more detailed study of the influence of spatial gradients and boundary conditions. A convenient formalism, which is a generalization of the formalism of [2], is used in this study. The following has been shown for a Boltzmann gas (compare [1–4]):

1. a)
   The Boltzmann equation is applicable for describing flows in which the condition of molecular chaos is satisfied and in which the characteristic dimension L (time T) is much greater than the diameter d (time τ_c) of molecular interactions.     
   
   

Numerical and asymptotic solutions of generalised Burgers’ equation

The generalised Burgers’ equation models the nonlinear evolution of acoustic disturbances subject to thermoviscous dissipation. When thermoviscous effects are small, asymptotic analysis predicts the development of a narrow shock region, which widens, leading eventually to a shock-free linear decay regime. The exact nature of the evolution differs subtly depending upon whether plane waves are considered, or cylindrical or spherical spreading waves. This paper focuses on the differences in asymptotic shock structure and validates the asymptotic predictions by comparison with numerical solutions. Precise expressions for the shock width and shock location are also obtained.     

Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods

A new family of Monte Carlo schemes has been recently introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics (SIAM J. Sci. Comput. 2001; 23 :1253–1273). After a splitting of the equation the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high‐order terms in the expansion with the equilibrium Maxwellian distribution. The corresponding time relaxed Monte Carlo (TRMC) schemes allow the use of time steps larger than those required by direct simulation Monte Carlo (DSMC) and guarantee consistency in the fluid‐limit with the compressible Euler equations. Conservation of mass, momentum, and energy are also preserved by the schemes. Applications to a two‐dimensional gas dynamic flow around an obstacle are presented which show the improvement in terms of computational efficiency of TRMC schemes over standard DSMC for regimes close to the fluid‐limit. Copyright © 2005 John Wiley & Sons, Ltd.     

Exact solutions of model BGK Boltzmann equation in temperature jump and weak evaporation problems

An exact solution to the model Boltzmann equation with Bhatnagar-Gross-Krook (BGK) collision operator is obtained in the problems of weak evaporation and temperature and density jumps of a rarefied gas in a half-space. Case's method is used to find generalized eigenvectors of the corresponding characteristic equation. An existence and uniqueness theorem for the solution of the posed problems with boundary conditions on a flat surface and far from it is proved. For this, we develop a formalism of diagonalization and factorization of the vector Riemann-Hilbert boundary-value problem with matrix coefficient whose diagonalizing matrix has branch points in the complex plane.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–171, January–February, 1992.     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

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Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

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Work supported by the C.N.R. through the G.N.F.M.     

On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off

We consider a Boltzmann gas which fills all of space and is under the influence of a field of conservative external force whose potential is bounded from below.Assuming the intermolecular force has a cut-off, we prove existence and uniqueness for the general solution of the nonlinear Maxwell-Boltzmann equation at least in a finite interval of time. The solution can be constructed by the method of successive approximations in corresponding complete spaces. Definitions of these spaces are connected with an exponential function of the total energy of the molecule.Some indications of future generalizations and investigations are given.