From hyperbolic systems to kinetic theory : a personalized quest / Luc Tartar.

Format:

Book

Author/Creator:

Tartar, Luc.

Series:

Lecture notes of the Unione Matematica Italiana ; 6.

Lecture notes of the Unione Matematica Italiana, 1862-9113 ; 6

Language:

English

Subjects (All):

Continuum mechanics.

Differential equations, Hyperbolic.

Kinetic theory of gases.

Dynamics.

Mathematical physics.

Physical Description:

1 online resource (306 p.)

Edition:

1st ed. 2008.

Place of Publication:

Berlin : Springer, 2008.

Language Note:

English

Summary:

Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!

Contents:

Historical Perspective

Hyperbolic Systems: Riemann Invariants, Rarefaction Waves

Hyperbolic Systems: Contact Discontinuities, Shocks

The Burgers Equation and the 1-D Scalar Case

The 1-D Scalar Case: the E-Conditions of Lax and of Oleinik

Hopf's Formulation of the E-Condition of Oleinik

The Burgers Equation: Special Solutions

The Burgers Equation: Small Perturbations; the Heat Equation

Fourier Transform; the Asymptotic Behaviour for the Heat Equation

Radon Measures; the Law of Large Numbers

A 1-D Model with Characteristic Speed 1/?

A 2-D Generalization; the Perron–Frobenius Theory

A General Finite-Dimensional Model with Characteristic Speed 1/?

Discrete Velocity Models

The Mimura–Nishida and the Crandall–Tartar Existence Theorems

Systems Satisfying My Condition (S)

Asymptotic Estimates for the Broadwell and the Carleman Models

Oscillating Solutions; the 2-D Broadwell Model

Oscillating Solutions: the Carleman Model

The Carleman Model: Asymptotic Behaviour

Oscillating Solutions: the Broadwell Model

Generalized Invariant Regions; the Varadhan Estimate

Questioning Physics; from Classical Particles to Balance Laws

Balance Laws; What Are Forces?

D. Bernoulli: from Masslets and Springs to the 1-D Wave Equation

Cauchy: from Masslets and Springs to 2-D Linearized Elasticity

The Two-Body Problem

The Boltzmann Equation

The Illner–Shinbrot and the Hamdache Existence Theorems

The Hilbert Expansion

Compactness by Integration

Wave Front Sets; H-Measures

H-Measures and “Idealized Particles”

Variants of H-Measures

Biographical Information

Abbreviations and Mathematical Notation.

Notes:

"ISSN electronic edition 1862-9121."

Includes bibliographical references and index.

ISBN:

1-281-23173-8

9786611231736

3-540-77562-5

OCLC:

233973813