Scaling Limits and Models in Physical Processes / Edition 1by Carlo Cercignani, David Sattinger
Pub. Date: 09/01/1998
Publisher: Birkhauser Basel
This is an introductory text, in two parts, on scaling limits and modelling in equations of mathematical physics. The first part is concerned with basic concepts of the kinetic theory of gases which is not only important in its own right but also as a prototype of a mathematical construct central to the theory of non-equilibrium phenomena in large systems. It also… See more details below
This is an introductory text, in two parts, on scaling limits and modelling in equations of mathematical physics. The first part is concerned with basic concepts of the kinetic theory of gases which is not only important in its own right but also as a prototype of a mathematical construct central to the theory of non-equilibrium phenomena in large systems. It also features a very readable historic survey of the field. The second part dwells on the role of integrable systems for modelling weakly nonlinear equations which contain the effects of both dispersion and nonlinearity. Starting with a historical introduction to the subject and a description of numerical techniques, it proceeds to a discussion of the derivation of the Korteweg de Vries and nonlinear Schrödinger equations, followed by a careful treatment of the inverse scattering theory for the Schrödinger operator. The book provides an up-to-date and detailed overview to this very active area of research and is intended as an accessible introduction for non-specialists and graduate students in mathematics, physics and engineering.
Table of Contents
I 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.
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