299 G(t), and to obtain the corresponding properties of its Laplace transform (called the resolvent of - A) R(p) = (A + pl)-l , whose existence is linked with the spectrum of A. The functional space framework used will be, for simplicity, a Banach space(3). To summarise, we wish to extend definition (2) for bounded operators A, i.e. G(t) = exp( - tA) , to unbounded operators A over X, where X is now a Banach space. Plan of the Chapter We shall see in this chapter that this enterprise is possible, that it gives us in addition to what is demanded above, some supplementary information in a number of areas: - a new 'explicit' expression of the solution; - the regularity of the solution taking into account some conditions on the given data (u , u1,f etc ... ) with the notion of a strong solution; o - asymptotic properties of the solutions. In order to treat these problems we go through the following stages: in § 1, we shall study the principal properties of operators of semigroups {G(t)} acting in the space X, particularly the existence of an upper exponential bound (in t) of the norm of G(t). In §2, we shall study the functions u E X for which t --+ G(t)u is differentiable.

Table of Contents:
XIV. Evolution Problems: Cauchy Problems in IRn.- §1. The Ordinary Cauchy Problems in Finite Dimensional Spaces.- 1. Linear Systems with Constant Coefficients.- 2. Linear Systems with Non Constant Coefficients.- §2. Diffusion Equations.- 1. Setting of Problem.- 2. The Method of the Fourier Transform.- 3. The Elementary Solution of the Heat Equation.- 4. Mathematical Properties of the Elementary Solution and the Semigroup Associated with the Heat Operator.- §3. Wave Equations.- 1. Model Problem: The Wave Equation in ?n.- 2. The Euler—Poisson—Darboux Equation.- 3. An Application of §2 and 3: Viscoelasticity.- §4. The Cauchy Problem for the Schrödinger Equation, Introduction.- 1. Model Problem 1. The Case of Zero Potential.- 2. Model Problem 2. The Case of a Harmonic Oscillator.- §5. The Cauchy Problem for Evolution Equations Related to Convolution Products.- 1. Setting of Problem.- 2. The Method of the Fourier Transform.- 3. The Dirac Equation for a Free Particle.- §6. An Abstract Cauchy Problem. Ovsyannikov’s Theorem.- Review of Chapter XIV.- XV. Evolution Problems: The Method of Diagonalisation.- §1. The Fourier Method or the Method of Diagonalisation.- 1. The Case of the Space ?1(n = 1).- 2. The Case of Space Dimension n = 2.- 3. The Case of Arbitrary Dimension n.- Review.- §2. Variations. The Method of Diagonalisation for an Operator Having Continuous Spectrum.- 1. Review of Self-Adjoint Operators in Hilbert Spaces.- 2. General Formulation of the Problem.- 3. A Simple Example of the Problem with Continuous Spectrum.- §3. Examples of Application: The Diffusion Equation.- 1. Example of Application 1: The Monokinetic Diffusion Equation for Neutrons.- 2. Example of Application 2: The Age Equation in Problems of Slowing Down of Neutrons.- 3. Example of Application 3: Heat Conduction.- §4. The Wave Equation: Mathematical Examples and Examples of Application.- 1. The Case of Dimension n = 1.- 2. The Case of Arbitrary Dimension n.- 3. Examples of Applications for n = 1.- 4. Examples of Applications for n = 2. Vibrating Membranes.- 5. Application to Elasticity; the Dynamics of Thin Homogeneous Beams.- §5. The Schrödinger Equation.- 1. The Cauchy Problem for the Schrödinger Equation in a Domain ? = ]0, 1[? ?.- 2. A Harmonic Oscillator.- Review.- §6. Application with an Operator Having a Continuous Spectrum: Example.- Review of Chapter XV.- Appendix. Return to the Problem of Vibrating Strings.- XVI. Evolution Problems: The Method of the Laplace Transform.- §1. Laplace Transform of Distributions.- 1. Study of the Set If and Definition of the Laplace Transform.- 2. Properties of the Laplace Transform.- 3. Characterisation of Laplace Transforms of Distributions of L+ (?).- §2. Laplace Transform of Vector-valued Distributions.- 1. Distributions with Vector-valued Values.- 2. Fourier and Laplace Transforms of Vector-valued Distributions.- §3. Applications to First Order Evolution Problems.- 1. ‘Vector-valued Distribution’ Solutions of an Evolution Equation of First Order in t.- 2. The Method of Transposition.- 3. Application to First Order Evolution Equations. The Hilbert Space Case. L2 Solutions in Hilbert Space.- 4. The Case where A is Defined by a Sesquilinear Form a(u, v).- §4. Evolution Problems of Second Order in t.- 1. Direct Method.- 2. Use of Symbolic Calculus.- Review.- §5. Applications.- 1. Hydrodynamical Problems.- 2. A Problem of the Kinetics of Neutron Diffusion.- 3. Problems of Diffusion of an Electromagnetic Wave.- 4. Problems of Wave Propagation.- 5. Viscoelastic Problems.- 6. A Problem Related to the Schrödinger Equation.- 7. A Problem Related to Causality, Analyticity and Dispersion Relations.- 8. Remark 10.- Review of Chapter XVI.- XVII. Evolution Problems: The Method of Semigroups.- A. Study of Semigroups.- §1. Definitions and Properties of Semigroups Acting in a Banach Space.- 1. Definition of a Semigroup of Class &0 (Resp. of a Group).- 2. Basic Properties of Semigroups of Class &0.- §2. The Infinitesimal Generator of a Semigroup.- 1. Examples.- 2. The Infinitesimal Generator of a Semigroup of Class &0.- §3. The Hille—Yosida Theorem.- 1. A Necessary Condition.- 2. The Hille—Yosida Theorem.- 3. Examples of Application of the Hille—Yosida Theorem.- §4. The Case of Groups of Class &0 and Stone’s Theorem.- 1. The Characterisation of the Infinitesimal Generator of a Group of Class &0.- 2. Unitary Groups of Class &0. Stone’s Theorem.- 3. Applications of Stone’s Theorem.- 4. Conservative Operators and Isometric Semigroups in Hilbert Space.- Review.- §5. Differentiable Semigroups.- §6. Holomorphic Semigroups.- §7. Compact Semigroups.- 1. Definition and Principal Properties.- 2. Characterisation of Compact Semigroups.- 3. Examples of Compact Semigroups.- B. Cauchy Problems and Semigroups.- §1. Cauchy Problems.- §2. Asymptotic Behaviour of Solutions as t ? + ?. Conservation and Dissipation in Evolution Equations.- §3. Semigroups and Diffusion Problems.- §4. Groups and Evolution Equations.- 1. Wave Problems.- 2. Schrödinger Type Problems.- 3. Weak Asymptotic Behaviour, for t ? ± ?, of Solutions of Wave Type of Schrödinger Type Problems.- 4. The Cauchy Problem for Maxwell’s Equations in an Open Set ? ? ?3.- §5. Evolution Operators in Quantum Physics. The Liouville—von Neumann Equation.- 1. Existence and Uniqueness of the Solution of the Cauchy Problem for the Liouville—von Neumann Equation in the Space of Trace Operators.- 2. The Evolution Equation of (Bounded) Observables in the Heisenberg Representation.- 3. Spectrum and Resolvent of the Operator h.- §6. Trotter’s Approximation Theorem.- 1. Convergence of Semigroups.- 2. General Representation Theorem.- Summary of Chapter XVII.- XVIII. Evolution Problems: Variational Methods.- Orientation.- §1. Some Elements of Functional Analysis.- 1. Review of Vector-valued Distributions.- 2. The Space W(a, b; V, V’).- 3. The Spaces W(a, b; X, Y).- 4. Extension to Banach Space Framework.- 5. An Intermediate Derivatives Theorem.- 6. Bidual. Reflexivity. Weak Convergence and Weak * Convergence.- §2. Galerkin Approximation of a Hilbert Space.- 1. Definition.- 2. Examples.- 3. The Outline of a Galerkin Method.- §3. Evolution Problems of First Order in t.- 1. Formulation of Problem (P).- 2. Uniqueness of the Solution of Problem (P).- 3. Existence of a Solution of Problem (P).- 4. Continuity with Respect to the Data.- 5. Appendix: Various Extensions — Liftings.- §4. Problems of First Order in t (Examples).- 1. Mathematical Example 1. Dirichlet Boundary Conditions.- 2. Mathematical Example 2. Neumann Boundary Conditions.- 3. Mathematical Example 3. Mixed Dirichlet—Neumann Boundary Conditions.- 4. Mathematical Example 4. Bilinear Form Depending on Time t.- 5. Evolution, Positivity and ‘Maximum’ of Solutions of Diffusion Equations in Lp(?), 1 ? p ? ?.- 6. Mathematical Example 5. A Problem of Oblique Derivatives.- 7. Example of Application. The Neutron Diffusion Equation.- 8. A Stability Result.- §5. Evolution Problems of Second Order in t.- 1. General Formulation of Problem (P1).- 2. Uniqueness in Problem (P1).- 3. Existence of a Solution of Problem (P1).- 4. Continuity with Respect to the Data.- 5. Formulation of Problem (P2).- §6. Problems of Second Order in t. Examples.- 1. Mathematical Example 1.- 2. Mathematical Example 2.- 3. Mathematical Example 3.- 4. Mathematical Example 4.- 5. Application Examples.- §7. Other Types of Equation.- 1. Schrödinger Type Equations.- 2. Evolution Equations with Delay.- 3. Some Integro-Differential Equations.- 4. Optimal Control and Problems where the Unknowns are Operators.- 5. The Problem of Coupled Parabolic-Hyperbolic Transmission.- 6. The Method of ‘Extension with Respect to a Parameter’.- Review of Chapter XVIII.- Table of Notations.- of Volumes 1–4, 6.