Beyer H.R. Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

Springer, 2007. — 283 p.This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.Conventions.
Mathematical Introduction.
Quantum Theory.
Wave Equations.
Prerequisites.
Linear Operators in Banach Spaces.
Weak Integration of Banach Space-Valued Maps.
Exponentials of Bounded Linear Operators.
Strongly Continuous Semigroups.
Elementary Properties.
Characterizations.
An Integral Representation in the Complex Case.
Perturbation Theorems.
Strongly Continuous Groups.
Associated Inhomogeneous Initial Value Problems.
Examples of Generators of Strongly Continuous Semigroups.
The Ordinary Derivative on a Bounded Interval.
Linear Stability of Ideal Rotating Couette Flows.
Outgoing Boundary Conditions.
DampedWave Equations.
Autonomous Linear Hermitian Hyperbolic Systems.
Intertwining Relations, Operator Homomorphisms.
Semigroups and Their Restrictions.
Intertwining Relations.
Nonexpansive Homomorphisms.
Examples of Constrained Systems.
1-D Wave Equations with Sommerfeld Boundary Conditions.
1-D Wave Equations with Engquist-Majda Boundary Conditions.
Maxwell’s Equations in Flat Space.
Kernels, Chains, and Evolution Operators.
A Convolution Calculus with Operator-Valued Kernels.
Chains.
Juxtaposition of Chains.
Finitely Generated Chains.
Evolution Operators.
Stable Families of Generators.
The Linear Evolution Equation.
Examples of Linear Evolution Equations.
Scalar Fields in the Gravitational Field of a Spherical Black Hole.
Non-Autonomous Linear Hermitian Hyperbolic Systems.
The Quasi-Linear Evolution Equation.
Examples of Quasi-Linear Evolution Equations.
A Generalized Inviscid Burgers’ Equation.
Quasi-Linear Hermitian Hyperbolic Systems.