This book considers evolution equations of hyperbolic and parabolic type. These equations are studied from a common point of view, using elementary methods, such as that of energy estimates, which prove to be quite versatile. The authors emphasize the Cauchy problem and present a unified theory for the treatment of these equations. In particular, they provide local and global existence results, as well as strong well-posedness and asymptotic behavior results for the Cauchy problem for quasi-linear equations. Solutions of linear equations are constructed explicitly, using the Galerkin method; the linear theory is then applied to quasi-linear equations, by means of a linearization and fixed-point technique. The authors also compare hyperbolic and parabolic problems, both in terms of singular perturbations, on compact time intervals, and asymptotically, in terms of the diffusion phenomenon, with new results on decay estimates for strong solutions of homogeneous quasi-linear equations of each type.
This textbook presents a valuable introduction to topics in the theory of evolution equations, suitable for advanced graduate students. The exposition is largely self-contained. The initial chapter reviews the essential material from functional analysis. New ideas are introduced along with their context. Proofs are detailed and carefully presented. The book concludes with a chapter on applications of the theory to Maxwell's equations and von Karman's equations.
Readership: Graduate students and research mathematicians interested in partial differential equations.
Author(s): Pascal Cherrier, Albert Milani
Series: Graduate Studies in Mathematics Vol. 135
Publisher: American Mathematical Society
Year: 2012
Language: English
Pages: C+xviii+377+B
Cover
S Title
Linear and Quasi-linear Evolution Equations in Hilbert Spaces, GSM 135
Copyright
© 2012 by the American Mathematical Society
ISBN 978-0-8218-7576-6
QA378.C44 2012 515'.733-dc23
LCCN 2012002958
Dedication
Contents
Preface
Chapter 1 Functional Framework
1.1. Basic Notation
1. Intervals, Balls, Integer Part
2. Derivatives
3. Spaces of Continuously Differentiable Functions
4. Integrals
5. Convolution
6. Conjugate Indices.
7. Constants
1.2. Functional Analysis Results
1. Imbeddings
2. Duality, Weak Convergence.
3. Bases.
1.3. Holder Spaces
1. Holder Spaces in \Omega
2. Holder Spaces in Q.
1.4. Lebesgue Spaces
1. Spaces L^p
2. Regularization
3. Inequalities
4. Interpolation of Lebesgue Spaces
1.5. Sobolev Spaces
1.5.1. Definitions and Main Properties
1. Spaces H^m(\Omega) and H^m(\Omega)
2. Spaces Hs(R^N).
3. Traces
4. Extensions and Restrictions
5. Regularization
6. Gagliardo-Nirenberg Inequalities
7. Interpolation of Sobolev Spaces
8. Imbedding Properties
9. Sobolev Product Estimates.
10. Spaces Vm
1.5.2. The Laplace Operator
1.5.3. Chain Rules and Commutator Estimates
1.5.4. Mollifiers and Commutator Estimates
1.6. Orthogonal Bases in H^m
1. Hermite Functions
2. A Basis of H.
1.7. Sobolev Spaces Involving Time
1. Bochner Space
2. Continuity and Differentiability.
3. Regularization
4. Spaces W
5. Intermediate Derivatives and Traces
6. Imbeddings.
Chapter 2 Linear Equations
2.1. Introduction
2.2. The Hyperbolic Cauchy Problem
2.3. Proof of Theorem 2.2.1
2.3.1. A Priori Estimates and Well-Posedne
2.3.2. Existence of Strong Solutions
2.3.3. Additional Regularity
2.4. Weak Solutions
2.5. The Parabolic Cauchy Problem
2.5.1. Strong Solutions
2.5.2. Regularity for t > 0.
2.5.3. Sobolev and Holder Solution
Chapter 3 Quasi-linear Equations
3.1. Introduction
3.2. The Hyperbolic Cauchy Problem
3.2.1. Strong Solutions
3.2.2. Preliminary Lemmas
3.2.3. Linear Estimates
3.3. Proof of Theorem 3.2.1
3.3.1. Step 1: Linearization.
3.3.2. Step 2: Contractivity
3.3.3. Step 3: Lipschitz Estimates
3.3.4. Step 4: Strong Well-Posedness
3.3.5. Step 5: Regularity
3.4. The Parabolic Cauchy Problem
Chapter 4 Global Existence
4.1. Introduction
4.2. Life Span of Solutions
4.3. Non Dissipative Finite Time Blow-Up
4.3.1. Lax's Example.
4.3.2. Geometrical Interpretation.
4.3.3. Invariant Regions.
4.4. Almost Global Existence
4.5. Global Existence for Dissipative Equations
4.5.1. The Linear Dissipative Equation.
1. The Solution Kernel.
2. Linear Decay Estimates for the Homogeneous Equation
3. Bounded Solutions of the Non Homogeneous Equation
4. The Autonomous Case
5. Optimality of Decay Rates.
4.5.2. Bounded Global Existence.
4.5.3. Global Existence
4.5.4. Dissipative Finite Time Blow-Up
4.6. The Parabolic Problem
4.6.1. The Solution Kernel
4.6.2. Bounded Global Existence
4.6.3. Global Existence, I.
4.6.4. Regularity for t > 0.
4.6.5. Global Existence, H.
Chapter 5 Asymptotic Behavior
5.1. Introduction
5.2. Convergence u^hyp(t) --->u^sta
5.3. Convergence
5.4. Stability Estimates
5.4.1. Hyperbolic Decay
5.4.2. Parabolic Decay
5.5. The Diffusion Phenomenon
5.5.1. The Linear Case
5.5.2. The Quasi-Linear Case.
Chapter 6 Singular Convergence
6.1. Introduction
6.2. An Example from ODEs
6.3. Uniformly Local and Global Existence
6.4. Singular Perturbation
6.4.1. Singular Convergence
6.4.2. The Initial Layer.
6.4.3. Comparison of Solutions.
6.5. Almost Global Existence
Chapter 7 Maxwell and von Karman Equations
7.1. Maxwell's Equations
7.1.1. The Equations.
1. Physical Principles.
2. Potentials
7.1.2. Solution Theory.
1. Assumptions
2. Main Result.
7.2. von Karman's Equations
7.2.1. The Equations.
1. The operators
2. The Equations.
3. Basic Function Spaces
4. Properties of N and I.
5. Elliptic Type Estimates on f.
7.2.2. The Hyperbolic System
1. Local Existence
2. Higher Regularity.
3. Almost Global Existence
7.2.3. The Parabolic System.
1. Local Existence.
2. Proof of Theorem 7.2.4.
3. Higher Regularity
4. Almost Global Existence.
List of Function Spaces
Bibliography
Index
Back Cover
