Table Of Contents:
Introduction 1(2)

What is this book about? 3(10)

Uniform weak solutions for the Navier-Stokes equations 5(1)

Mild solutions 6(4)

Energy inequalities 10(3)
Part 1: Some results of real harmonic analysis 13(90)

Real interpolation, Lorentz spaces and Sobolev embeddings 15(8)

A primer to real interpolation theory 15(3)

Lorentz spaces 18(2)

Sobolev inequalities 20(3)

Besov spaces and Littlewood-Paley decomposition 23(8)

The Littlewood-Paley decomposition of tempered distributions 23(2)

Besov spaces as real interpolation spaces of potential spaces 25(3)

Homogeneous Besov spaces 28(3)

Shift-invariant Banach spaces of distributions and related Besov spaces 31(8)

Shift-invariant Banach spaces of distributions 31(3)

Besov spaces 34(1)

Homogeneous spaces 35(4)

Vector-valued integrals 39(8)

The case of Lebesgue spaces 39(2)

Spaces Lp(E) 41(3)

Heat kernel and Besov spaces 44(3)

Complex interpolation, Hardy space and Calderon-Zygmund operators 47(10)

The Marcinkiewicz interpolation theorem and the Hardy-Littlewood maximal function 47(3)

The complex method in interpolation theory 50(1)

Atomic Hardy space and Calderon-Zygmund operators 51(6)

Vector-valued singular integrals 57(10)

Calderon-Zygmund operators 57(5)

Littlewood-Paley decomposition in Lp 62(2)

Maximal Lp(Lq) regularity for the heat kernel 64(3)

A primer to wavelets 67(12)

Multiresolution analysis 68(5)

Daubechies wavelets 73(4)

Multivariate wavelets 77(2)

Wavelets and functional spaces 79(12)

Lebesgue spaces 79(2)

Besov spaces 81(7)

Singular integrals 88(3)

The space BMO 91(12)

Carleson measures and the duality between H1 and BMO 91(4)

The T(1) theorem 95(5)

The local Hardy space h1 and the local space BMO 100(3)
Part 2: A general framework for shift-invariant estimates for the Navier-Stokes equations 103(30)

Weak solutions for the Navier-Stokes equations 105(10)

The Leray projection operator and the Oseen kernel 105(2)

Elimination of the pressure 107(5)

Differential formulation and the integral formulation for the Navier-Stokes equations 112(3)

Divergence-free vector wavelets 115(8)

A short survey in divergence-free vector wavelets 115(1)

Bi-orthogonal bases 116(4)

The div-curl theorem 120(3)

The mollified Navier-Stokes equations 123(10)

The mollified equations 123(5)

The limiting process 128(2)

Mild solutions 130(3)
Part 3: Classical existence results for the Navier-Stoke equations 133(24)

The Leray solutions for the Navier-Stokes equations 135(10)

The energy inequality 135(4)

Energy equality 139(3)

Uniqueness theorems 142(3)

The Kato theory of mild solutions for the Navier-Stokes equations 145(12)

Picard's contraction principle 145(3)

Kato's mild solutions in Hs, s ≥ d/2 - 1 148(3)

Kato's mild solutions in Lp, p ≥ d 151(6)
Part 4: New approaches to mild solutions 157(88)

The mild solutions of Koch and Tataru 159(12)

The space BMO-1 159(3)

Local and global existence of solutions 162(5)

Fourier transform, Navier-Stokes and BMO(-1) 167(4)

Generalization of the Lp theory: Navier-Stokes and local measures 171(8)

Shift-invariant spaces of local measures 171(2)

Kato's theorem for local measures: the direct approach 173(2)

Kato's theorem for local measures: the role of B-1 ∞, ∞ 175(4)

Further results for local measures 179(10)

The role of the Morrey-Campanato space M1,d and of bmo(-1) 179(2)

A persistency theorem 181(2)

Some alternate proofs for the existence of global solutions 183(6)

Regular initial values 189(8)

Cannone's adapted spaces 189(3)

Sobolev spaces and Besov spaces of positive order 192(2)

Persistency results 194(3)

Besov spaces of negative order 197(8)

Lp(Lq) solutions 197(3)

Potential spaces and Besov spaces 200(2)

Persistency results 202(3)

Pointwise multipliers of negative order 205(16)

Multipliers and Morrey-Campanato spaces 205(6)

Solutions in Xr 211(4)

Perturbated Navier-Stokes equations 215(6)

Further adapted spaces for the Navier-Stokes equations 221(12)

The analysis of Meyer and Muschietti 221(5)

The case of Besov spaces of null regularity 226(1)

The analysis of Auscher and Tchamitchian 226(7)

Cannone's approach of self similarity 233(12)

Besov spaces 233(6)

The Lorentz space Ld,∞ 239(2)

Asymptotic self-similarity 241(4)
Part 5: Decay and regularity results for weak and mild solutions 245(70)

Solutions of the Navier-Stokes equations are space-analytical 247(8)

The Le Jan and Sznitman solutions 247(2)

Analyticity of solutions in Hd/2-1 249(1)

Analyticity of solutions in Ld 250(5)

Space localization and Navier-Stokes equations 255(12)

The molecules of Furioli and Terraneo 255(5)

Spatial decay of velocities 260(4)

Vorticities are well localized 264(3)

Time decay for the solutions to the Navier-Stokes equations 267(10)

Wiegner's fundamental lemma and Schonbek's Fourier splitting device 267(1)

Decay rates for the L2 norm 268(4)

Optimal decay rate for the L2 norm 272(5)

Uniqueness of Ld solutions 277(12)

The uniqueness problem. 277(2)

Uniqueness in Ld 279(6)

The case of Morrey-Campanato spaces 285(4)

Further results on uniqueness of mild solutions 289(14)

Nonboundedness of the bilinear operator B on C([0,T], (Ld)d) 289(2)

Uniqueness in L∞ (Ld) (d ≥ 4) 291(2)

A uniqueness result in B-1 &ifin;, ∞ 293(10)

Stability and Lyapunov functionals 303(12)

Stability in Lebesgue norms 303(5)

A new Bernstein inequality 308(1)

Stability and Besov norms 309(6)
Part 6: Local energy inequalities for the Navier-Stokes equations on R3 315(58)

The Caffarelli, Kohn, and Nirenberg regularity criterion 317(14)

Suitable solutions 317(5)

A fundamental inequality 322(2)

The regularity criterion 324(7)

On the dimension of the set of singular points 331(10)

Singular times 331(1)

Hausdorff dimension of the set of singularities for a suitable solution 332(2)

The second regularity criterion of Caffarelli, Kohn, and Nirenberg 334(7)

Local existence (in time) of suitable local square-integrable weak solutions 341(12)

Size estimates for u →ε 342(4)

Local existence of solutions 346(2)

Decay estimates for suitable solutions 348(5)

Global existence of suitable local square-integrable weak solutions 353(10)

Regularity of uniformly locally L2 suitable solutions 353(1)

A generalized Von Wahl uniqueness theorem 354(6)

Global existence of uniformly locally L2 suitable solutions 360(3)

Leray's conjecture on self similar singularities 363(10)

Hopf's strong maximum principle 363(1)

The C0 self similar Leray solutions are equal to 0 364(3)

The case of local control 367(6)
Conclusion 373(8)

Singular initial values 375(6)

Allowed initial values 375(1)

Maximal regularity and critical spaces 376(1)

Mixed initial values 377(4)
References 381(2)
Bibliography 383(8)
Author index 391(2)
Subject index 393