Linear and quasi-linear evolution equations in Hilbert spaces

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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


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