Fourier analysis and nonlinear partial differential equations

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Fourier analysis and nonlinear partial differential equations / Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin

資料種別:

図書

出版情報:

Heidelberg : Springer, c2011

形態:

xv, 523 p. ; 25 cm

シリーズ名:

Die Grundlehren der mathematischen Wissenschaften ; 343 <BA00007422>

著者名:

ISBN:

9783642168291 [3642168299] (: hbk)
9783642266577 [3642266576] (: pbk)

書誌ID:

BB04695233

子書誌情報

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書誌詳細

注記:

Includes bibliographical references (p. 497-515) and index

主題:

Differential equations, Nonlinear; Differential equations, Partial; Fourier analysis

言語:

英語

目次情報:

Basic Analysis / 1：

Basic Real Anslysis / 1.1：

Holder and Convolution Inequslities / 1.1.1：

The Atomic Decomposition / 1.1.2：

Proof of Refined Young Inequslityp8 / 1.1.3：

A Bilinear Interpolation Theorem / 1.1.4：

A Linear Interpolation Result / 1.1.5：

The Hardy-Littlewood Maximal Function / 1.1.6：

The Fourier Transform / 1.2：

Fourier Transforms of Functions and the Schwartz Space / 1.2.1：

Tempered Distributions and the Fourier Transform / 1.2.2：

A Few Calculations of Fourier Transforms / 1.2.3：

Homogeneous Sobolev Spaces / 1.3：

Definition and Basic Properties / 1.3.1：

Sobolev Embedding in Lebesgue Spaces / 1.3.2：

The Limit Case H^d/2 / 1.3.3：

The Embedding Theorem in Hölder Spaces / 1.3.4：

Nonhomogeneous Sobolev Spaces on R^d / 1.4：

Embedding / 1.4.1：

A Density Theorem / 1.4.3：

Hardy Inequality / 1.4.4：

References and Remarks / 1.5：

Littlewood-Paley Theory / 2：

Functions with Compactly Supported Fourier Transforms / 2.1：

Bernstein-Type Lemmas / 2.1.1：

The Smoothing Effect of Heat Flow / 2.1.2：

The Action of a Diffeomorphism / 2.1.3：

The Effects of Some Nonlinear Functions / 2.1.4：

Dyadic Partition of Unity / 2.2：

Homogeneous Besov Spaces / 2.3：

Characterizations of Homogeneous Besov Spaces / 2.4：

Besov Spaces, Lebesgue Spaces, and Refined Inequalities / 2.5：

Homogeneous Paradifferential Calculus / 2.6：

Homogeneous Bony Decomposition / 2.6.1：

Action of Smooth Functions / 2.6.2：

Time-Space Besov Spaces / 2.6.3：

Nonhomogeneous Besov Spaces / 2.7：

Nonhomogeneous Paradifferential Calculus / 2.8：

The Bony Decomposition / 2.8.1：

The Paralinearization Theorem / 2.8.2：

Besov Spaces and Compact Embeddings / 2.9：

Commutator Estimates / 2.10：

Around the Space B_{&infty;,&infty;}¹ / 2.11：

Transport and Transport-Diffusion Equations / 2.12：

Ordinary Differential Equations / 3.1：

The Cauchy-Lipschitz Theorem Revisited / 3.1.1：

Estimates for the Flow / 3.1.2：

A Blow-up Criterion for Ordinary Differential Equations / 3.1.3：

Transport Equations: The Lipschitz Case / 3.2：

A Priori Estimates in General Besov Spaces / 3.2.1：

Refined Estimates in Besov Spaces with Index 0 / 3.2.2：

Solving the Transport Equation in Besov Spaces / 3.2.3：

Application to a Shallow Water Equation / 3.2.4：

Losing Estimates for Transport Equations / 3.3：

Linear Loss of Regularity in Besov Spaces / 3.3.1：

The Exponential Loss / 3.3.2：

Limited Loss of Regularity / 3.3.3：

A Few Applications / 3.3.4：

Transport-Diffusion Equations / 3.4：

A Priori Estimates / 3.4.1：

Exponential Decay / 3.4.2：

Quasilinear Symmetric Systems / 3.5：

Definition and Examples / 4.1：

Linear Symmetric Systems / 4.2：

The Well-posedness of Linear Symmetric Systems / 4.2.1：

Finite Propagation Speed / 4.2.2：

Further Well-posedness Results for Linear Symmetric Systems / 4.2.3：

The Resolution of Quasilinear Symmetric Systems / 4.3：

Paralinearization and Energy Estimates / 4.3.1：

Convergence of the Scheme / 4.3.2：

Completion of the Proof of Existence / 4.3.3：

Uniqueness and Continuation Criterion / 4.3.4：

Data with Critical Regularity and Blow-up Criteria / 4.4：

Critical Besov Regularity / 4.4.1：

A Refined Blow-up Crndition / 4.4.2：

Continuity of the Flow Map / 4.5：

The Incompressible Navier-Stokes System / 4.6：

Basic Facts Concerning the Navier-Stokes System / 5.1：

Well-posedness in Sobolev Spaces / 5.2：

A General Result / 5.2.1：

The Behavior of the H^d/2-1 Norm Near 0 / 5.2.2：

Results Related to the Structure of the System / 5.3：

The Particular Case of Dimension Two / 5.3.1：

The Case of Dimension Three / 5.3.2：

An Elementary L^p Approach / 5.4：

The Endpoint Space for Picard's Scheme / 5.5：

The Use of the L¹-smoothing Effect of the Heat Flow / 5.6：

The Cannone-Meyer-Planchon Theorem Revisited / 5.6.1：

The Flow of the Solutions of the Navier-Stokes System / 5.6.2：

Anisotropic Viscosity / 5.7：

The Case of L² Data with One Vertical Derivative in L² / 6.1：

A Global Existence Result in Anisotropic Besov Spaces / 6.2：

Anisotropic Localization in Fourier Space / 6.2.1：

The Functional Framework / 6.2.2：

Statement of the Main Result / 6.2.3：

Some Technical Lemmas / 6.2.4：

The Proof of Existence / 6.3：

The Proof of Uniqueness / 6.4：

Euler System for Perfect Incompressible Fluids / 6.5：

Local Well-posedness Results for Inviscid Fluids / 7.1：

The Biot-Savart Law / 7.1.1：

Estimates for the Pressure / 7.1.2：

Another Formulation of the Euler System / 7.1.3：

Local Existence of Smooth Solutions / 7.1.4：

Uniqueness / 7.1.5：

Continuation Criteria / 7.1.6：

Global Existence Results in Dimension Two / 7.2：

Smooth Solutions / 7.2.1：

The Borderline Case / 7.2.2：

The Yudovich Theorem / 7.2.3：

The Inviscid Limit / 7.3：

Regularity Results for the Navier-Stokes System / 7.3.1：

The Smooth Case / 7.3.2：

The Rough Case / 7.3.3：

Viscous Vortex Patches / 7.4：

Results Related to Striated Regularity / 7.4.1：

A Stationary Estimate for the Velocity Field / 7.4.2：

Uniform Estimates for Striated Regularity / 7.4.3：

A Global Convergence Result for Striated Regularity / 7.4.4：

Application to Smooth Vortex Patches / 7.4.5：

Strichartz Estimates and Applications to Semilinear Dispersive Equations / 7.5：

Examples of Dispersive Estimates / 8.1：

The Dispersive Estimate for the Free Transport Equation / 8.1.1：

The Dispersive Estimates for the Schrdillger Equation / 8.1.2：

Integral of Oscillating Functions / 8.1.3：

Dispersive Estimates for the Wave Equation / 8.1.4：

The L² Boundedness of Some Fourier Integral Operators / 8.1.5：

Billnear Methods / 8.2：

The Duality Method and the TT* Argument / 8.2.1：

Strichartz Estimates: The Case q > 2 / 8.2.2：

Strichartz Estimates: The Endpoint Case q = 2 / 8.2.3：

Application to the Cubic Semilinear Schrödinger Equation / 8.2.4：

Strichartz Estimates for the Wave Equation / 8.3：

The Basic Strichartz Estimate / 8.3.1：

The Refined Strichartz Estimate / 8.3.2：

The Qulntic Wave Equation in R³ / 8.4：

The Cubic Wave Equation in R³ / 8.5：

Solutions in H¹ / 8.5.1：

Local and Global Well-posedness for Rough Data / 8.5.2：

The Nonlinear Interpolation Method / 8.5.3：

Application to a Class of Semilinear Wave Equations / 8.6：

Smoothing Effect in Quasilinear Wave Equations / 8.7：

A Well-posedness Result Based on an Energy Method / 9.1：

The Main Statement and the Strategy of its Proof / 9.2：

Refined Paralinearization of the Wave Equation / 9.3：

Reduction to a Microlocal Strichartz Estimate / 9.4：

Microlocal Strichartz Estimates / 9.5：

A Rather General Statement / 9.5.1：

Geometrical Optics / 9.5.2：

The Solution of the Eikonal Equation / 9.5.3：

The Transport Equation / 9.5.4：

The Approximation Theorem / 9.5.5：

The Proof of Theorem 9.16 / 9.5.6：

The Compressible Navier-Stokes System / 9.6：

About the Model / 10.1：

General Overview / 10.1.1：

The Barotropic Navier-Stokes Equations / 10.1.2：

Local Theory for Data with Critical Regularity / 10.2：

Scaling Invariance and Statement of the Main Result / 10.2.1：

Existence of a Local Solution / 10.2.2：

A Continuation Criterion / 10.2.4：

Local Theory for Data Bounded Away from the Vacuum / 10.3：

A Priori Estimates for the Linearized Momentum Equation / 10.3.1：

Global Existence for Small Data / 10.3.2：

Statement of the Results / 10.4.1：

A Spectral Analysis of the Linearized Equation / 10.4.2：

A Prioli Estimates for the Linearized Equation / 10.4.3：

Proof of Global Existence / 10.4.4：

The Incompressible Limit / 10.5：

Main Results / 10.5.1：

The Case of Small Data with Critical Regularity / 10.5.2：

The Case of Large Data with More Regularity / 10.5.3：

References / 10.6：

List of Notations

Index

Basic Analysis / 1：

Basic Real Anslysis / 1.1：

Holder and Convolution Inequslities / 1.1.1：

The Atomic Decomposition / 1.1.2：

Proof of Refined Young Inequslityp8 / 1.1.3：

A Bilinear Interpolation Theorem / 1.1.4：

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