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 Hd/2 / 1.3.3: |
The Embedding Theorem in Hölder Spaces / 1.3.4: |
Nonhomogeneous Sobolev Spaces on Rd / 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 Hd/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 Lp 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: |