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＜E-Book＞
Mathematical Analysis and Numerical Methods for Science and Technology : Volume 5 Evolution Problems I / by Robert Dautray, Jacques-Louis Lions

Edition	1st ed. 2000.
Publisher	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
Year	2000
Size	XIV, 739 p : online resource
Authors	*Dautray, Robert author Lions, Jacques-Louis author SpringerLink (Online service)
Subjects	LCSH:Partial differential equations LCSH:Numerical analysis FREE:Partial Differential Equations FREE:Numerical Analysis
Notes	XIV. Evolution Problems: Cauchy Problems in IRn -- §1. The Ordinary Cauchy Problems in Finite Dimensional Spaces -- §2. Diffusion Equations -- §3. Wave Equations -- §4. The Cauchy Problem for the Schrödinger Equation, Introduction -- §5. The Cauchy Problem for Evolution Equations Related to Convolution Products -- §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 -- §2. Variations. The Method of Diagonalisation for an Operator Having Continuous Spectrum -- §3. Examples of Application: The Diffusion Equation -- §4. The Wave Equation: Mathematical Examples and Examples of Application -- §5. The Schrödinger Equation -- §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 -- §2. Laplace Transform of Vector-valued Distributions -- §3. Applications to First Order Evolution Problems -- §4. Evolution Problems of Second Order in t -- §5. Applications -- 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 -- §2. The Infinitesimal Generator of a Semigroup -- §3. The Hille—Yosida Theorem -- §4. The Case of Groups of Class &0 and Stone’s Theorem -- §5. Differentiable Semigroups -- §6. Holomorphic Semigroups -- §7. 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 -- §5. Evolution Operators in Quantum Physics. The Liouville—von Neumann Equation -- §6. Trotter’s Approximation Theorem -- Summary of Chapter XVII -- XVIII. Evolution Problems: Variational Methods -- Orientation -- §1. Some Elements of Functional Analysis -- §2. Galerkin Approximation of a Hilbert Space -- §3. Evolution Problems of First Order in t -- §4. Problems of First Order in t (Examples) -- §5. Evolution Problems of Second Order in t -- §6. Problems of Second Order in t. Examples -- §7. Other Types of Equation -- Review of Chapter XVIII -- Table of Notations -- of Volumes 1–4, 6 299 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 HTTP:URL=https://doi.org/10.1007/978-3-642-58090-1

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