On diffuse reflection at the boundary for the Boltzmann equation and related equations

The paper considers diffuse reflection at the boundary with nonconstant boundary temperature and unbounded velocities. The solutions obtained are proved to conserve mass at the boundary. After a preliminary study of the collisionless case, the main results obtained are existence for the Boltzmann equation in a DiPerna-Lions framework with the above boundary conditions in a bounded measure sense, and existence together with uniqueness for the BGK equation with Maxwellian diffusion on the boundary in anL framework.Deceased.     

Small data existence for the Enskog equation inL 1

An existence theorem for the Enskog equation with small initial data is proved in anL ¹ setting. This type of result is not available for the Boltzmann equation.     

On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation

This paper considers the linear space-inhomogeneous Boltzmann equation for a distribution function in a bounded domain with general boundary conditions together with an external potential force. The paper gives results on strong convergence to equilibrium, whent, for general initial data; first in the cutoff case, and then for infinite-range collision forces. The proofs are based on the properties of translation continuity and weak convergence to equilibrium. To handle these problems generalH-theorems (concerning monotonicity in time of convex entropy functionals) are presented. Furthermore, the paper gives general results on collision invariants, i.e., on functions satisfying detailed balance relations in a binary collision.     

Biased lattice gases with correlated equilibrium states

The approach to and structure of the equilibrium state is studied for a 7-bit lattice gas with biased forward and backward transition rates by means of mean field theory and computer simulations. If the rate constants obey the factorizability and the detailed balance conditions, the occupations of different velocity directions are uncorrelated, anH-theorem is valid, and a nonuniversal equilibrium state exists that depends explicitly on the transition rates. In case the above conditions are not satisfied, theH-theorem is no longer valid, and mean field theory also predicts nontrivial velocity correlations in postcollision states. The simulations are mainly concentrated on the time dependence of pre- and postcollision velocity correlations on a single node, and on slowly increasing fluctuations that might indicate metastable behavior.     

H-theorem for the (modified) nonlinear Enskog equation

We construct anH-function suitable for a system of dense hard spheres satisfying the (modified) nonlinear Enskog equation and we show that _t H 0. The equality sign holds only when the system has reached absolute equilibrium, in which caseS=– k_B H becomes the exact equilibrium entropy of the hard-sphere fluid.     

Global existence inL 1 for the generalized Enskog equation

Various existence theorems are given for the generalized Enskog equation inR ³ and in a bounded spatial domain with periodic boundary conditions. A very general form of the geometric factorY is allowed, including an explicit space, velocity, and time dependence. The method is based on the existence of a Liapunov functional, an analog of theH-function in the Boltzmann equation, and utilizes a weak compactness argument inL ¹.     

The L p Spectrum of Locally Symmetric Spaces with Small Fundamental Group

We determine the L ^p spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L ^p spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L ² case, the restriction on the injectivity radius is not needed.      

Existence and uniqueness for nonlinear boundary value problems in kinetic theory

A boundary value problem for the stationary nonlinear Boltzmann equation in a slab has been examined in a weightedL space. It has been proved that the problem possesses a unique solution for boundary data small enough. The proof is based on the implicit function theorem. It has also been shown that for the linearized problem the Fredholm alternative applies.     

Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at ; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.Reference due to C. Cercignani.     

General H-Theorem for Hard Spheres

The maximum entropy formalism is used to investigate the growth of entropy (H-theorem) for an isolated system of hard spheres in an external potential under general boundary geometry. Assuming that only correlations of a finite number of particles are controlled and the rest maximizes entropy, we obtain an H-theorem for such a system The limiting cases such as the modified Enskog equation and linear kinetic theory are discussed.     

On Isotropic Distributional Solutions to the Boltzmann Equation for Bose-Einstein Particles

The paper considers the spatially homogeneous Boltzmann equation for Bose-Einstein particles (BBE). In order to include the hard sphere model, the equation is studied in a weak form and its solutions (including initial data) are set in the class of isotropic positive Borel measures and therefore called isotropic distributional solutions. Stability of distributional solutions is established in the weak topology, global existence of distributional solutions that conserve the mass and energy is proved by weak convergence of approximate L ¹-solutions, and moment production estimates for the distributional solutions are also obtained. As an application of the weak form of the BBE equation, it is shown that a Bose-Einstein distribution plus a Dirac dt-function is an equilibrium solution to the BBE equation in the weak form if and only if it satisfies a low temperature condition and an exact ratio of the Bose-Einstein condensation.     

LocalH-theorem for a kinetic variational theory

A localH-theorem is derived for a recently proposed extension of Enskog kinetic theory to a dense model fluid composed of particles with interactions extending beyond a hard core.On leave from: Katedra Fizyki, Uniwersytetu Szczecinskiego, 70-451 Szczecin, Poland.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

Integral Representation of the Linear Boltzmann Operator for Granular Gas Dynamics with Applications

We investigate the properties of the collision operator Q associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of Q in an Hilbert space setting, generalizing results from T. Carleman (Publications Scientifiques de l’Institut Mittag-Leffler, vol. 2, 1957) to granular gases. In the same way, we obtain from this integral representation of Q ⁺ that the semigroup in L ¹(ℝ³×ℝ³,dx ⊗ dv) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from Arlotti (Acta Appl. Math. 23:129–144, 1991).     

On the Initial Boundary Value Problems for the Enskog Equation in Irregular Domains

The paper is concerned with the Enskog equation with a constant high density factor for large initial data in L ¹(R ⁿ). The initial boundary value problem is investigated for bounded domains with irregular boundaries. The proof of an H-theorem for the case of general domains and boundary conditions is given. The main result guarantees the existence of global solutions of boundary value problems for large initial data with all v-moments initially finite and domains having boundary with finite Hausdorff measure and satisfying a cone condition. Existence and uniqueness are first proved for the case of bounded velocities. The solution has finite norm where q = (t ₀, x) is taken on all possible n-dimensional planes Q(v) in R ^n+l intersecting a fixed point and orthogonal to vectors (1, v), v R ⁿ.     

Inviscid Limits¶of the Complex Ginzburg–Landau Equation

In the inviscid limit the generalized complex Ginzburg–Landau equation reduces to the nonlinear Schr?dinger equation. This limit is proved rigorously with H ¹ data in the whole space for the Cauchy problem and in the torus with periodic boundary conditions. The results are valid for nonlinearities with an arbitrary growth exponent in the defocusing case and with a subcritical or critical growth exponent at the level of L ² in the focusing case, in any spatial dimension. Furthermore, optimal convergence rates are proved. The proofs are based on estimates of the Schr?dinger energy functional and on Gagliardo–Nirenberg inequalities. Received: 2 April 1999 / Accepted: 29 March 2000     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL ¹ sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andp ^th velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.On leave from School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332.On leave from C.F.M.C. and Departamento de Matemática da Faculdade de Ciencias de Lisboa, 1700 Lisboa codex, Portugal.     

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.     

Transition prediction for supersonic and hypersonic boundary layers on a cone with angle of attack

Transition prediction for boundary layers has always been one of the urgent problems waiting for a solution for the development of aero-space technology, yet there is no reliable and effective method due to the complexity of the problem. The e ^N method has been regarded as an effective method for the transition prediction of boundary layers. However, it heavily relies on experiment or experience. And in cases with three-dimensional base flow, for instance, the boundary layer on a cone with angle of attack, the result of its application is not satisfactory. The authors have found its cause and proposed the method for its improvement, which did yield the fairly satisfactory result for a given test case, and also did not rely so much on experiment or experience. However, before people can really apply this method to practical problems, more test cases have to be studied. In this paper, more test cases for the application of the improved e ^N method to problems of transition prediction of supersonic and hypersonic boundary layers on cones with angle of attack will be studied. The results are compared with those obtained by experiments and/or direct numerical simulations, confirming that the improved e ^N method is effective and reliable. We also find that there may be more than one ZARF for each meridian plane, and which one should be chosen for the e ^N method has been clarified. Supported by the National Natural Science Foundation of China (Grant Nos. 10632050 and 90716007), the Special Foundation for the Authors of National Excellent Doctoral Dissertations (Grant No. 200328), and the Foundation of Liu-Hui Center of Applied Mathematics of Nankai University and Tianjin University Contributed by ZHOU Heng