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Lecture Notes Of The Unione Matematica Italiana #6: From Hyperbolic Systems to Kinetic Theory: A Personalized Quest

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Synopses & Reviews

Publisher Comments:

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!

Synopsis:

This fascinating book, penned by Luc Tartar of America's Carnegie Mellon University, starts from the premise that equations of state are not always effective in continuum mechanics. Tartar relies on H-measures, a tool created for homogenization, to explain some of the weaknesses in the theory. These include looking at the subject from the point of view of quantum mechanics. Here, there are no "particles", so the Boltzmann equation and the second principle, can't apply.

Table of Contents

1.Historical Perspective.- 2.Hyperbolic Systems: Riemann Invariants, Rarefaction Waves.- 3.Hyperbolic Systems: Contact Discontinuities, Shocks.- 4.The Burgers Equation and the 1-D Scalar Case.- 5.The 1-D Scalar Case: the E-Conditions of Lax and of Oleinik.- 6.Hopf's Formulation of the E-Condition of Oleinik.- 7.The Burgers Equation: Special Solutions.- 8.The Burgers Equation: Small Perturbations; the Heat Equation.- 9.Fourier Transform; the Asymptotic Behaviour for the Heat Equation.- 10.Radon Measures; the Law of Large Numbers.- 11.A 1-D Model with Characteristic Speed 1/epsilon.- 12.A 2-D Generalization; the Perron-Frobenius Theory.- 13.A General Finite-Dimensional Model with Characteristic Speed 1/epsilon.- 14.Discrete Velocity Models.- 15.The Mimura-Nishida and the Crandall-Tartar Existence Theorems.- 16.Systems Satisfying My Condition (S).- 17.Asymptotic Estimates for the Broadwell and the Carleman Models.- 18.Oscillating Solutions; the 2-D Broadwell Model.- 19.Oscillating Solutions: the Carleman Model.- 20.The Carleman Model: Asymptotic Behaviour.- 21.Oscillating Solutions: the Broadwell Model.- 22.Generalized Invariant Regions; the Varadhan Estimate.- 23.Questioning Physics; from Classical Particles to Balance Laws.- 24.Balance Laws; What Are Forces?- 25.D. Bernoulli: from Masslets and Springs to the 1-D Wave Equation.- 26.Cauchy: from Masslets and Springs to 2-D Linearized Elasticity.- 27.The Two-Body Problem.- 28.The Boltzmann Equation.- 29.The Illner-Shinbrot and the Hamdache Existence Theorems.- 30.The Hilbert Expansion.- 31.Compactness by Integration.- 32.Wave Front Sets; H-Measures.- 33.H-Measures and "Idealized Particles".- 34.Variants of H-Measures.- 35.Biographical Information.- 36.Abbreviations and Mathematical Notation.- References.- Index.

Product Details

ISBN:
9783540775614
Author:
Tartar, Luc
Publisher:
Springer
Location:
Berlin, Heidelberg
Subject:
Dynamics
Subject:
Mathematical Physics
Subject:
Differential Equations
Subject:
Mechanics - General
Subject:
35K05, 35L45, 35L60, 35L65, 35L67, 35Q30, 70F45, 76A02, 76N1
Subject:
Navier-Stokes equation
Subject:
Conservation laws
Subject:
Fluid mechanics
Subject:
Heat equation.
Subject:
hyperbolic systems
Subject:
interacting particle systems
Subject:
PARTIAL DIFFERENTIAL EQUATIONS
Subject:
Classical Continuum Physics
Subject:
mathematical methods in physics
Subject:
Dynamical Systems and Ergodic Theory
Subject:
Mathematics-Differential Equations
Subject:
Mathematics
Subject:
Language, literature and biography
Subject:
mathematics and statistics
Subject:
Differential equations, partial
Subject:
Differentiable dynamical systems
Copyright:
Edition Description:
Book
Series:
Lecture Notes Of The Unione Matematica Italiana
Series Volume:
06
Publication Date:
20080410
Binding:
TRADE PAPER
Language:
English
Pages:
310
Dimensions:
235 x 155 mm

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Lecture Notes Of The Unione Matematica Italiana #6: From Hyperbolic Systems to Kinetic Theory: A Personalized Quest New Trade Paper
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Product details 310 pages Springer - English 9783540775614 Reviews:
"Synopsis" by , This fascinating book, penned by Luc Tartar of America's Carnegie Mellon University, starts from the premise that equations of state are not always effective in continuum mechanics. Tartar relies on H-measures, a tool created for homogenization, to explain some of the weaknesses in the theory. These include looking at the subject from the point of view of quantum mechanics. Here, there are no "particles", so the Boltzmann equation and the second principle, can't apply.
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