Singularly Perturbed Evolution Equations with Applications to Kinetic Theory

The following sections are included:

• Preface

• Contents

Since the times of Newton, physical phenomena were being described in terms of partial differential equations with time as one of the independent variables. Some 50 years ago such equations acquired the collective name evolution equations and a comprehensive theory of semigroups of operators related to solutions of evolution equations has been developed…

The following sections are included:

• Introduction

• General definitions and notation

• Banach and Hilbert spaces

• Distributions and Sobolev spaces

• Vector-valued functions

• Unbounded operators

• Elements of spectral theory

The following sections are included:

• Introduction

• Generation of semigroups

• Fractional powers of closed operators

• Perturbation theorems

• Asymptotic behaviour of solutions

• Inhomogeneous Cauchy problem

• Applications to partial differential equations

The following sections are included:

• Introduction

• Single evolution equations with a small parameter

• Systems of evolution equations with a small parameter

The following sections are included:

• Singular-singularly perturbed evolution equations

• Model system: exact solution

• Model system: standard asymptotic analysis

• Model system: compressed asymptotic expansion

• Modified model system

• Singular-singularly perturbed evolution equations: compressed approach

• Singularly perturbed linear kinetic equations

The following sections are included:

• Introduction

• Preliminary results

• Properties of terms of expansion

• Estimates of error of asymptotic expansion

• Remarks on non-selfadjointness of C

The following sections are included:

• Introduction

• Properties of linear Boltzmann equation with unbounded velocity range

• Diffusion approximation to linear Boltzmann equation

The following sections are included:

• Introduction

• General assumptions

• Variational setting

• Fokker-Planck equation of electron scattering in plasma

• Fokker-Planck equation of Brownian motion

• Fokker-Planck equation of vibrational relaxation of harmonic oscillator

The following sections are included:

• Introduction

• Properties of neutron transport equation in slab geometry

• Asymptotic expansion

• Remarks on general spatially inhomogeneous equation

The following sections are included:

• Introduction

• Properties of collision operator

• Formal asymptotic formulae

• Initial layer part

• Bulk part

• Error of asymptotic expansion

The following sections are included:

• Introduction

• Asymptotic analysis of telegraph systems

• Remarks on compressed asymptotic method in L₁-setting

• Carleman model

The following sections are included:

• Bibliography

• Index