From Hyperbolic Systems to Kinetic Theory: A Personalized Quest

Luc Tartar (auth.)3540775617, 978-3-540-77561-4

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!

Table of contents :
Front Matter….Pages I-XXVII
Historical Perspective….Pages 1-15
Hyperbolic Systems: Riemann Invariants, Rarefaction Waves….Pages 17-30
Hyperbolic Systems: Contact Discontinuities, Shocks….Pages 31-38
The Burgers Equation and the 1-D Scalar Case….Pages 39-44
The 1-D Scalar Case: the E-Conditions of Lax and of Oleinik….Pages 45-50
Hopf’s Formulation of the E-Condition of Oleinik….Pages 51-56
The Burgers Equation: Special Solutions….Pages 57-62
The Burgers Equation: Small Perturbations; the Heat Equation….Pages 63-71
Fourier Transform; the Asymptotic Behaviour for the Heat Equation….Pages 73-82
Radon Measures; the Law of Large Numbers….Pages 83-89
A 1-D Model with Characteristic Speed 1/ε….Pages 91-95
A 2-D Generalization; the Perron–Frobenius Theory….Pages 97-104
A General Finite-Dimensional Model with Characteristic Speed 1/ε….Pages 105-112
Discrete Velocity Models….Pages 113-128
The Mimura–Nishida and the Crandall–Tartar Existence Theorems….Pages 129-133
Systems Satisfying My Condition (S)….Pages 135-141
Asymptotic Estimates for the Broadwell and the Carleman Models….Pages 143-147
Oscillating Solutions; the 2-D Broadwell Model….Pages 149-156
Oscillating Solutions: the Carleman Model….Pages 157-162
The Carleman Model: Asymptotic Behaviour….Pages 163-168
Oscillating Solutions: the Broadwell Model….Pages 169-178
Generalized Invariant Regions; the Varadhan Estimate….Pages 179-185
Questioning Physics; from Classical Particles to Balance Laws….Pages 187-196
Balance Laws; What Are Forces?….Pages 197-200
D. Bernoulli: from Masslets and Springs to the 1-D Wave Equation….Pages 201-208
Cauchy: from Masslets and Springs to 2-D Linearized Elasticity….Pages 209-212
The Two-Body Problem….Pages 213-217
The Boltzmann Equation….Pages 219-227
The Illner–Shinbrot and the Hamdache Existence Theorems….Pages 229-232
The Hilbert Expansion….Pages 233-237
Compactness by Integration….Pages 239-244
Wave Front Sets; H-Measures….Pages 245-250
H-Measures and “Idealized Particles”….Pages 251-255
Variants of H-Measures….Pages 257-266
Biographical Information….Pages 267-270
Abbreviations and Mathematical Notation….Pages 271-274
Back Matter….Pages 275-280