Evolution problems

Author(s)

• Dautray, Robert
• Lions, Jacques Louis
• Artola, Michel
• Cessenat, M.
• Lanchon, Hélène
• Craig, Alan
• Sneddon, Ian Naismith
• Bardos, Claude

Bibliographic Information

Evolution problems

Robert Dautray, Jacques-Louis Lions ; with the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon ; translated from the French by Alan Craig ; translation editor, Ian N. Sneddon

（Mathematical analysis and numerical methods for science and technology / Robert Dautray, Jacques-Louis Lions, v. 5-6）

Springer-Verlag, 2000

• 1
• 2

Other Title

Analyse mathématique et calcul numérique pour les sciences et les techniques

Note

2: With the collaboration of Claude Bardos ...[et al.]

Includes indexes

Description and Table of Contents

Volume

1 ISBN 9783540661016

Description

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 Schroedinger 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 Schroedinger Equation.- 1. The Cauchy Problem for the Schroedinger 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 Schroedinger 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. Schroedinger Type Problems.- 3. Weak Asymptotic Behaviour, for t ? +/- ?, of Solutions of Wave Type of Schroedinger 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. Schroedinger 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.

Volume

2 ISBN 9783540661023

Description

These six volumes--the result of a ten year collaboration between two distinguished international figures--compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. It is a comprehensive and up-to-date publication that presents the mathematical tools needed in applications of mathematics.

Table of Contents

XIX. The Linearised Navier-Stokes Equations.- 1. The Stationary Navier-Stokes Equations: The Linear Case.- 1. Functional Spaces.- 2. Existence and Uniqueness Theorem.- 3. The Problem of L? Regularity.- 2. The Evolutionary Navier-Stokes Equations: The Linear Case.- 1. Functional Spaces and Trace Theorems.- 2. Existence and Uniqueness Theorem.- 3. L2-Regularity Result.- 3. Additional Results and Review.- 1. The Variational Approach.- 2. The Functional Approach.- 3. The Problem of L? Regularity for the Evolutionary Navier-Stokes Equations: The Linearised Case.- XX. Numerical Methods for Evolution Problems.- 1. General Points.- 1. Discretisation in Space and Time.- 2. Convergence, Consistency and Stability.- 3. Equivalence Theorem.- 4. Comments.- 5. Schemes with Constant Coefficients and Step Size.- 6. The Symbol of a Difference Scheme.- 7. The von Neumann Stability Condition.- 8. The Kreiss Stability Condition.- 9. The Case of Multilevel Schemes.- 10. Characterisation of a Scheme of Order q.- 2. Problems of First Order in Time.- 1. Introduction.- 2. Model Equation for x ? ?.- 3. The Boundary Value Problem for Equation.- 4. Equation with Variable Coefficients and Schemes with Variable Step-Size.- 5. The Heat Flow Equation in Two Space Dimensions.- 6. Alternating Direction and Fractional Step Methods.- 7. Internal Approximation Schemes.- 8. Integration of Systems of Stiff Differential Equations.- 9. Comments.- 3. Problems of Second Order in Time.- 1. Introduction.- 2. The Model Equation for x ? ?.- 3. The Wave Equation in Two Space Dimensions.- 4. Internal Approximation Schemes.- 5. The Newmark Scheme.- 6. The Wave Equation with Viscosity.- 7. The Wave Equation Coupled to a Heat Flow Equation.- 8. Comments.- 4. The Advection Equation.- 1. Introduction.- 2. Some Explicit Schemes for the Cauchy Problem in One Space Dimension.- 3. Positive-Type Schemes and Stable Schemes in LX(?).- 4. Some Explicit Schemes.- 5. The Problem with Boundary Conditions.- 6. Phase and Amplitude Error. Schemes of Order Greater than Two.- 7. Nonlinear Schemes for the Equation.- 8. Difference Schemes for the Cauchy Problem with Many Space Variables.- 5. Symmetric Friedrichs Systems.- 1. Introduction.- 2. Summary of Symmetric Friedrichs Systems.- 3. Finite Difference Schemes for the Cauchy Problem.- 4. Approximation of Boundary Conditions in the Case where ? = ]0, 1 [.- 5. Maxwell's Equations.- 6. Remarks.- 6. The Transport Equation.- 1. Introduction.- 2. Stationary Equation in One-Dimensional Plane Geometry.- 3. The Evolution Equation in One-Dimensional Plane Geometry.- 4. The Equation in One-Dimensional Spherical Geometry.- 5. Iterative Solution of Schemes Approximating the Transport Equation.- 6. The Two-Dimensional Equation.- 7. Other Methods.- 8. Comments.- 7. Numerical Solution of the Stokes Problem.- 1. Setting of Problem.- 2. An Integral Method.- 3. Some Finite Difference Methods.- 4. Finite Element Methods.- 5. Some Methods Using the Stream function.- 6. The Evolutionary Stokes Problem.- XXI. Transport.- 1. Introduction. Presentation of Physical Problems.- 1. Evolution Problems in Neutron Transport.- 2. Stationary Problems.- 3. Principal Notation.- 2. Existence and Uniqueness of Solutions of the Transport Equation.- 1. Introduction.- 2. Study of the Advection Operator A = - v. ?.- 3. Solution of the Cauchy Transport Problem.- 4. Solution of the Stationary Transport Problem in the Subcritical Case.- Summary.- Appendix of 2. Boundary Conditions in Transport Problems. Reflection Conditions.- 3. Spectral Theory and Asymptotic Behaviour of the Solutions of Evolution Problems.- 1. Introduction.- 2. Study of the Spectrum of the Operator B = - v. ? - ?.- 3. Study of the Spectrum of the Transport Operator in an Open Bounded Set X of ?n.- 4. Positivity Properties.- 5. The Particular Case where All the Eigenvalues are Real.- 6. The Spectrum of the Transport Operator in a Band. The Lehner-Wing Theorem.- 7. Study of the Spectrum of the Transport Operator in the Whole Space: X = ?n.- 8. The Spectrum of the Transport Operator on the Exterior of an "Obstacle".- 9. Some Remarks on the Spectrum of T.- Summary.- Appendix of 3. The Conservative Milne Problem.- 4. Explicit Examples.- 1. The Stationary Transport Problem in the Whole Space ?.- 2. The Evolutionary Transport Problem in the Whole Space.- 3. The Stationary Transport Problem in the Half-Space by the Method of "Invariant Embedding".- 4. Case's Method of "Generalised Eigenfunctions". Application to the Critical Dimension in the Case of a Band.- 5. Approximation of the Neutron Transport Equation by the Diffusion Equation.- 1. Physical Introduction.- 2. Approximation in the Case of a Monokinetic Model of Evolution Equations and of Stationary Transport Equations.- 3. Generalisation of Section 2.- 4. Calculation of a Corrector for the Stationary Problem and Extrapolation Length.- 5. Convergence of the Principal Eigenvalue of the Transport Operator.- 6. Calculation of a Corrector for the Principal Eigenvalue of the Transport Operator.- 7. Application to a Critical Size Problem.- 8. Numerical Example in the Case of a Band.- Appendix of 5.- Perspectives.- Orientation for the Reader.- List of Equations.- Table of Notations.- Cumulative Index of Volumes 1-6.- of Volumes 1-5.

by "Nielsen BookData"

Details

• NCID: BA44533273
• ISBN:
  • 3540661018
  • 3540661026
• Country Code: gw
• Title Language Code: eng
• Text Language Code: eng
• Original Language Code: fre
• Place of Publication: Berlin
• Pages/Volumes: 2 v.
• Size: 24 cm
• Subject Headings:
  • LCSH : Mathematical analysis
  • LCSH : Numerical analysis
• Parent Bibliography ID:
  • BA0456824X