Detailansicht
Scaling Limits and Models in Physical Processes
Oberwolfach Seminars 28
ISBN/EAN: 9783764359850
Umbreit-Nr.: 604029
Sprache:
Englisch
Umfang: vi, 194 S., 2 s/w Illustr., 194 p. 2 illus.
Format in cm: 1.5 x 24 x 17
Einband:
kartoniertes Buch
Erschienen am 01.09.1998
- Zusatztext
- The first part of this volume presents the basic ideas concerning perturbation and scaling methods in the mathematical theory of dilute gases, based on Boltzmann's integro-differential equation. It is of course impossible to cover the developments of this subject in less than one hundred pages. Already in 1912 none less than David Hilbert indicated how to obtain approximate solutions of the scaled Boltzmann equation in the form of a perturbation of a parameter inversely proportional to the gas density. His paper is also reprinted as Chapter XXII of his treatise Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen. The motive for this circumstance is clearly stated in the preface to that book ("Recently I have added, to conclude, a new chapter on the kinetic theory of gases. [. ]. I recognize in the theory of gases the most splendid application of the theorems concerning integral equations. ") The mathematically rigorous theory started, however, in 1933 with a paper [48] by Tage Gillis Torsten Carleman, who proved a theorem of global exis tence and uniqueness for a gas of hard spheres in the so-called space-homogeneous case. Many other results followed; those based on perturbation and scaling meth ods will be dealt with in some detail. Here, I cannot refrain from mentioning that, when Pierre-Louis Lions obtained the Fields medal (1994), the commenda tion quoted explicitly his work with the late Ronald DiPerna on the existence of solutions of the Boltzmann equation.
- Autorenportrait
- InhaltsangabeI Scaling and Mathematical Models in Kinetic Theory.- 1 Boltzmann Equation and Gas Surface Interaction.- 1.1 Introduction.- 1.2 The Boltzmann equation.- 1.3 Molecules different from hard spheres.- 1.4 Collision invariants.- 1.5 The Boltzmann inequality and the Maxwell distributions.- 1.6 The macroscopic balance equations.- 1.7 The H-theorem.- 1.8 Equilibrium states and Maxwellian distributions.- 1.9 Model equations.- 1.10 Boundary conditions.- 2 Perturbation Methods for the Boltzmann Equation.- 2.1 Introduction.- 2.2 Rarefaction regimes.- 2.3 Solving the Boltzmann equation. Analytical techniques.- 2.4 Hydrodynamical limit and other scalings.- 2.5 The linearized collision operator.- 2.6 The basic properties of the linearized collision operator.- 2.7 Spectral properties of the Fourier-transformed, linearized Boltzmann equation.- 2.8 The asymptotic behavior of the solution of the Cauchy problem for the linearized Boltzmann equation.- 2.9 A quick survey of the global existence theorems for the nonlinear equation.- 2.10 Hydrodynamical limits. A formal discussion.- 2.11 The Hilbert expansion.- 2.12 The entropy approach to the hydrodynamical limit.- 2.13 The hydrodynamic limit for short times.- 2.14 Other scalings and the incompressible Navier-Stokes equations.- 2.15 Concluding remarks.- II Scaling, Mathematical Modelling, & Integrable Systems.- 1 Dispersion.- 1.1 Introduction.- 1.2 Group and phase velocities.- 2 Nonlinear Schrödinger Equation.- 2.1 Multiple scales expansion.- 2.2 Pulse solutions.- 3 Korteweg-de Vries.- 3.1 Background and history.- 3.2 Plasmas.- 3.3 Water waves.- 3.4 The solitary wave of the KdV equation.- 4 Isospectral Deformations.- 4.1 The KdV hierarchy.- 4.2 The AKNS hierarchy.- 5 Inverse Scattering Theory.- 5.1 The Schrödinger equation.- 5.2 First Order Systems.- 5.3 Decay of the scattering data.- 6 Variational Methods.- 6.1 Water Waves.- 6.2 Method of Averaging.- 7 Weak and Strong Nonlinearities.- 7.1 Breaking and Peaking.- 7.2 Strongly nonlinear models.- 7.3 The extended AKNS hierarchy.- 8 Numerical Methods.- 8.1 The finite Fourier transform.- 8.2 Pseudospectral codes.