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4 edition of Singular and nonuniform limits of solutions of the Boltzmann equation. found in the catalog.

Singular and nonuniform limits of solutions of the Boltzmann equation.

by Harold Grad

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  • 36 Currently reading

Published by Courant Institute of Mathematical Sciences, New York University in New York .
Written in English


The Physical Object
Pagination99 p.
Number of Pages99
ID Numbers
Open LibraryOL17868824M

On the asymptotics of the Boltzmann equation and fluid-dynamic limits Kazuo Aoki Department of Mechanical Engineering and Science, Kyoto University, Kyoto , Japan [email protected] We consider a rarefied gas in . The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up.

Linear algebraic formulations of discretized, mono-energetic, steady-state, linear Boltzmann transport equations (BTE) in three dimensions are presented. The discretizations consist of a discrete ordinates collocation in angle and a Petrov–Galerkin finite element method in space. A matrix development of diffusion synthetic acceleration (DSA) is given for three-dimensional (3-D) rectangular Cited by: PARTICLE HYDRODYNAMIC MOMENT MODELS IN BIOLOGY AND MICROELECTRONICS: SINGULAR RELAXATION LIMITS GUI-QIANG CHENa, JOSEPH W. JEROMEa and BO ZHANGb aDepartment of Mathematics, Northwestern University, Evanston, IL , USA; and bDepartment of Mathematics, Stanford University, Stanford, CA , USA Key words and phrases hydrodynamic model, reduced hydrodynamic model, relaxation limit.

scheme with a high spectral accuracy for smooth solutions. Deterministic methods for the Boltzmann equation have been extensively studied in the last decades. Overview of these methods can be found, for example, in the book of V. Aristov [2] and in a more recent review by A. Narayan and A. Kl¨ockner [34]. Since the pioneering work of D. these terms satisfy the Boltzmann equation order by order, partly due to the complicated selection rules. In this paper, we present a derivation of the quantum Boltzmann equation under the main assumption that in the weak coupling limit (8) the four-point and the .


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Singular and nonuniform limits of solutions of the Boltzmann equation by Harold Grad Download PDF EPUB FB2

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Publication date Publisher New York: Courant Institute of Mathematical Sciences, New York UniversityPages: singular and nonuniform limits of solutions of the boltzmann equation by harold grad at stcroixcaribbeanweddings.com - the best online ebook storage.

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Mar 29,  · A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules.

The fluid velocity u, rather than the spatial co-ordinate x, is used as the independent variable, Cited by: Boltzmann equation, the solution f Singular and nonuniform limits of solutions of This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation.

For celebrated Boltzmann equation from gas dynamics, which describes elastic interactions of particles, the fluid dynamics limit is, at some assumptions, just five Euler equations of hydrodynamics. This number of hydrodynamic equations arises due to five conservation laws for solutions of the Boltzmann equation of gas stcroixcaribbeanweddings.com: Pavel B.

Dubovski. To be exact, that fact made it possible for Maxwell to solve the problem, deriving the equation of continuous medium from a Boltzmann equation.

However, Chapman-Enskog solution for arbitrary potentials is essentially based on given solution for Maxwellian molecules. For Maxwell-Chapman-Enskog method, it is sufficient deal with first momentum equations.

The final solution of momentum system. The Boltzmann Equation and Its Hydrodynamic Limits in the book [28]. Hence, although not a first principle itself, the Boltzmann equation is rig- physical legitimacy than phe-nomenological models (such as lattice gases or stochastic Hamiltonian models). Besides, the Boltzmann equation is currently used by engineers in aerospace Cited by: It extends the convergence result for well-prepared initial data obtained by the author in [L.

Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit Author: Laure Saint-Raymond. List of Works Cited The following list, not a bibliography of the kinetic theory, is limited to works which seem to us to have made a contribution, if in some cases only peripheral, to the specific subject of this book.

n˙(r;k;t) evolve in time. To simplify matters, we will consider a single band and drop the indices n˙. It is clear that in the absence of collisions, the distribution function must satisfy the continuity equation, @f @t + r(uf) = 0: () This is just the condition of number conservation for electrons.

A rigorous theory of the Boltzmann equation, which has thus far been successful only for cutoff potentials, should be extended to cases of Infinite total cross-section. The principal purpose Is mathematical, viz., to find out whether such a very singular form of the Boltzmann equation Is a legitimate mathematical enterprise.

Boltzmann equation, in particular the position-space locality and the Stoßzahlansatz (IV)— and a “non-kinetic part” f, which should be irrelevant for weakly interacting systems.

This. QUANTITATIVE POINTWISE ESTIMATE OF THE SOLUTION OF THE LINEARIZED BOLTZMANN EQUATION YU-CHU LIN, HAITAO WANG, AND KUNG-CHIEN WU Abstract.

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad’s angular cuto assumption.

We prove some converse properties of long time limiting behavior (along the particle paths) of a class of spatial decay solutions of the Boltzmann equation.

It is shown that different initial data Cited by: Aug 01,  · The Knudsen layer in rarefied gas dynamics is essentially described by a half-space boundary-value problem of the linearized Boltzmann equation, in which the incoming data are specified on the boundary and the solution is assumed to be bounded at infinity (Milne problem).

This problem is considered for a binary mixture of hard-sphere gases, and the existence and uniqueness of the solution Cited by: The structure of normal shock waves is investigated on the basis of the standard Boltzmann equation for hard‐sphere molecules.

This fundamental nonlinear problem in rarefied gas dynamics is analyzed numerically by a newly developed finite‐difference method, where the Boltzmann collision integral is computed directly without using the Monte Carlo stcroixcaribbeanweddings.com by: We only obtain one solution because the cubic equation () degenerates to a linear equation at "= 0.

We missed the other two solutions because they approach in nity as "!0. A change in the qualitative nature of the problem at the unperturbed value "= 0 is typical of singular perturbation problems.

Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case.

As a further study on the problem in the spatially homogeneous Cited by:. In biophysics and certain surface chemistry applications, it is known simply as the Poisson–Boltzmann equation.

It is also known in electrochemistry as Gouy-Chapman theory; in solution chemistry as Debye–Huckel theory; in colloid chemistry as Derjaguin–Landau–Verwey–Overbeek (DLVO) theory.Cylindrical Couette flows of a rarefied gas between two concentric circular cylinders consisting of the condensed phase of the gas, where evaporation or condensation occurs, are considered on the basis of kinetic theory, with interest in the behavior of the gas in the continuum limit.

The limiting solution is obtained by asymptotic analysis of the Boltzmann stcroixcaribbeanweddings.com by: The Linear Boltzmann Equation 1. Introduction One must distinguish between the “linear Boltzmann equation” and the “linearized Boltzmann equation.” The former has no self interaction, just scattering with the medium, whereas the latter is the linearization of the fully nonlinear Boltzmann equation.

We will deal with the linear equation.