Para-differential calculus and applications to the Cauchy problem for nonlinear systems /
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| Author / Creator: | Métivier, Guy. |
|---|---|
| Imprint: | [Pisa, Italy] : Edizioni della Normale, c2008. |
| Description: | xi, 140 p. ; 24 cm. |
| Language: | English |
| Series: | CRM series ; 5 CRM series (Pisa, Italy) ; 5. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/7629653 |
Table of Contents:
- Preface
- Part I. Introduction to systems
- 1. Notations and examples
- 1.1. First order systems
- 1.1.1. Notations
- 1.1.2. Plane waves
- 1.1.3. The symbol
- 1.2. Examples
- 1.2.1. Gas dynamics
- 1.2.2. Maxwell's equations
- 1.2.3. Magneto-hydrodynamics
- 1.2.4. Elasticity
- 2. Constant coefficient systems. Fourier synthesis
- 2.1. The method
- 2.1.1. The Fourier transform
- 2.1.2. Solving the evolution equation (2.1)
- 2.2. Examples
- 2.2.1. The heat equation
- 2.2.2. Schrodinger equation
- 2.2.3. The wave equation
- 2.3. First order systems: hyperbolicity
- 2.3.1. The general formalism
- 2.3.2. Strongly hyperbolic systems
- 2.3.3. Symmetric hyperbolic systems
- 2.3.4. Smoothly diagonalizable systems, hyperbolic systems with constant multiplicities
- 2.3.5. Existence and uniqueness for strongly hyperbolic systems
- 2.4. Higher order systems
- 2.4.1. Systems of Schrodinger equations
- 2.4.2. Elasticity
- 3. The method of symmetrizers
- 3.1. The method
- 3.2. The constant coefficients case
- 3.2.1. Fourier multipliers
- 3.2.2. The first order case
- 3.3. Hyperbolic symmetric systems
- 3.3.1. Assumptions
- 3.3.2. Existence and uniqueness
- 3.3.3. Energy estimates
- Part II. The para-differential calculus
- 4. Pseudo-differential operators
- 4.1. Fourier analysis of functional spaces
- 4.1.1. Smoothing and approximation
- 4.1.2. The Littlewood-Paley decomposition in H[superscript s]
- 4.1.3. The Littlewood-Paley decomposition in Holder spaces
- 4.2. The general framework of pseudo-differential operators
- 4.2.1. Introduction
- 4.2.2. Operators with symbols in the Schwartz class
- 4.2.3. Pseudo-differential operators of type (1, 1)
- 4.2.4. Spectral localization
- 4.3. Action of pseudo-differential operators in Sobolev spaces
- 4.3.1. Stein's theorem for operators of type (1, 1)
- 4.3.2. The case of symbols satisfying spectral conditions
- 5. Para-differential operators
- 5.1. Definition of para-differential operators
- 5.1.1. Symbols with limited spatial smoothness
- 5.1.2. Smoothing symbols
- 5.1.3. Operators
- 5.2. Paraproducts
- 5.2.1. Definition
- 5.2.2. Products
- 5.2.3. Para-linearization 1
- 5.2.4. Para-linearization 2
- 6. Symbolic calculus
- 6.1. Composition
- 6.1.1. Statement of the result
- 6.1.2. Proof of the main theorem
- 6.1.3. A quantitative version
- 6.2. Adjoints
- 6.2.1. The main result
- 6.3. Applications
- 6.3.1. Elliptic estimates
- 6.3.2. Garding's inequality
- 6.4. Pluri-homogeneous calculus
- Part III. Applications
- 7. Nonlinear hyperbolic systems
- 7.1. The L[superscript 2] linear theory
- 7.1.1. Statement of the result
- 7.1.2. Paralinearisation
- 7.1.3. Symmetrizers
- 7.1.4. The basic L[superscript 2] estimate
- 7.1.5. Weak= Strong and uniqueness
- 7.1.6. Existence
- 7.2. The H[superscript s] linear theory
- 7.2.1. Statement of the result
- 7.2.2. Paralinearisation
- 7.2.3. Estimates
- 7.2.4. Smoothing effect in time
- 7.2.5. Existence
- 7.3. Quasi-linear systems
- 7.3.1. Statement of the results
- 7.3.2. Local in time existence
- 7.3.3. Blow up criterion
- 8. Systems of Schrodinger equations
- 8.1. Introduction
- 8.1.1. Decoupling
- 8.1.2. Further reduction
- 8.2. Energy estimates for linear systems
- 8.2.1. The results
- 8.2.2. Proof of Theorem 8.2.4
- 8.3. Existence, uniqueness and smoothness for linear problems
- 8.3.1. L[superscript 2] existence
- 8.3.2. H[superscript s] existence
- 8.4. Nonlinear problems
- 8.4.1. Systems with quasilinear first order part
- 8.4.2. Examples and applications
- References
