LEADER 04861nam 2200709 a 450 001 9910784744203321 005 20200520144314.0 010 $a1-281-23173-8 010 $a9786611231736 010 $a3-540-77562-5 024 7 $a10.1007/978-3-540-77562-1 035 $a(CKB)1000000000404196 035 $a(EBL)337065 035 $a(OCoLC)233973813 035 $a(SSID)ssj0000299187 035 $a(PQKBManifestationID)11196149 035 $a(PQKBTitleCode)TC0000299187 035 $a(PQKBWorkID)10240754 035 $a(PQKB)10552031 035 $a(DE-He213)978-3-540-77562-1 035 $a(MiAaPQ)EBC337065 035 $a(Au-PeEL)EBL337065 035 $a(CaPaEBR)ebr10223381 035 $a(CaONFJC)MIL123173 035 $a(PPN)123743346 035 $a(EXLCZ)991000000000404196 100 $a20081029d2008 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aFrom hyperbolic systems to kinetic theory$b[electronic resource] $ea personalized quest /$fLuc Tartar 205 $a1st ed. 2008. 210 $aBerlin $cSpringer$d2008 215 $a1 online resource (306 p.) 225 1 $aLecture notes of the Unione Matematica Italiana,$x1862-9113 ;$v6 300 $a"ISSN electronic edition 1862-9121." 311 $a3-540-77561-7 320 $aIncludes bibliographical references and index. 327 $aHistorical 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. 330 $aEquations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity! 410 0$aLecture notes of the Unione Matematica Italiana ;$v6. 606 $aContinuum mechanics 606 $aDifferential equations, Hyperbolic 606 $aKinetic theory of gases 606 $aDynamics 606 $aMathematical physics 615 0$aContinuum mechanics. 615 0$aDifferential equations, Hyperbolic. 615 0$aKinetic theory of gases. 615 0$aDynamics. 615 0$aMathematical physics. 676 $a531 700 $aTartar$b Luc$056396 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910784744203321 996 $aFrom hyperbolic systems to kinetic theory$91225179 997 $aUNINA