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Author Dautray, Robert. author Mathematical Analysis and Numerical Methods for Science and Technology [electronic resource] : Volume 5 Evolution Problems I / by Robert Dautray, Jacques-Louis Lions Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000 http://dx.doi.org/10.1007/978-3-642-58090-1 XIV, 739 p. online resource

SUMMARY

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 ยง 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 ยง2, we shall study the functions u E X for which t --+ G(t)u is differentiable

CONTENT

XIV. Evolution Problems: Cauchy Problems in IRn -- ยง1. The Ordinary Cauchy Problems in Finite Dimensional Spaces -- ยง2. Diffusion Equations -- ยง3. Wave Equations -- ยง4. The Cauchy Problem for the Schrรถdinger Equation, Introduction -- ยง5. The Cauchy Problem for Evolution Equations Related to Convolution Products -- ยง6. An Abstract Cauchy Problem. Ovsyannikovโ{128}{153}s Theorem -- Review of Chapter XIV -- XV. Evolution Problems: The Method of Diagonalisation -- ยง1. The Fourier Method or the Method of Diagonalisation -- ยง2. Variations. The Method of Diagonalisation for an Operator Having Continuous Spectrum -- ยง3. Examples of Application: The Diffusion Equation -- ยง4. The Wave Equation: Mathematical Examples and Examples of Application -- ยง5. The Schrรถdinger Equation -- ยง6. 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 -- ยง1. Laplace Transform of Distributions -- ยง2. Laplace Transform of Vector-valued Distributions -- ยง3. Applications to First Order Evolution Problems -- ยง4. Evolution Problems of Second Order in t -- ยง5. Applications -- Review of Chapter XVI -- XVII. Evolution Problems: The Method of Semigroups -- A. Study of Semigroups -- ยง1. Definitions and Properties of Semigroups Acting in a Banach Space -- ยง2. The Infinitesimal Generator of a Semigroup -- ยง3. The Hilleโ{128}{148}Yosida Theorem -- ยง4. The Case of Groups of Class &0 and Stoneโ{128}{153}s Theorem -- ยง5. Differentiable Semigroups -- ยง6. Holomorphic Semigroups -- ยง7. Compact Semigroups -- B. Cauchy Problems and Semigroups -- ยง1. Cauchy Problems -- ยง2. Asymptotic Behaviour of Solutions as t ? + ?. Conservation and Dissipation in Evolution Equations -- ยง3. Semigroups and Diffusion Problems -- ยง4. Groups and Evolution Equations -- ยง5. Evolution Operators in Quantum Physics. The Liouvilleโ{128}{148}von Neumann Equation -- ยง6. Trotterโ{128}{153}s Approximation Theorem -- Summary of Chapter XVII -- XVIII. Evolution Problems: Variational Methods -- Orientation -- ยง1. Some Elements of Functional Analysis -- ยง2. Galerkin Approximation of a Hilbert Space -- ยง3. Evolution Problems of First Order in t -- ยง4. Problems of First Order in t (Examples) -- ยง5. Evolution Problems of Second Order in t -- ยง6. Problems of Second Order in t. Examples -- ยง7. Other Types of Equation -- Review of Chapter XVIII -- Table of Notations -- of Volumes 1โ{128}{147}4, 6

Mathematics Partial differential equations Numerical analysis Mathematics Partial Differential Equations Numerical Analysis

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