The Navier-Stokes Problem in the 21st Century : Table of Contents

Presentation of the Clay Millennium Prizes

Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century
The Clay Millennium Prizes
The Clay Millennium Prize for the Navier–Stokes equations
Boundaries and the Navier–Stokes Clay Millennium Problem

The physical meaning of the Navier–Stokes equations

Frames of references
The convection theorem
Conservation of mass
Newton's second law
Pressure
Strain
Stress
The equations of hydrodynamics
The Navier–Stokes equations
Vorticity
Boundary terms
Blow up
Turbulence

History of the equation

Mechanics in the Scientific Revolution era
Bernoulli's Hydrodymica
D'Alembert
Euler
Laplacian physics
Navier, Cauchy, Poisson, Saint-Venant, and Stokes
Reynolds
Oseen, Leray, Hopf, and Ladyzhenskaya
Turbulence models

Classical solutions

The heat kernel
The Poisson equation
The Helmholtz decomposition
The Stokes equation
The Oseen tensor
Classical solutions for the Navier–Stokes problem
Small data and global solutions
Time asymptotics for global solutions
Steady solutions
Spatial asymptotics
Spatial asymptotics for the vorticity
Intermediate conclusion

A capacitary approach of the Navier–Stokes integral equations

The integral Navier–Stokes problem
Quadratic equations in Banach spaces
A capacitary approach of quadratic integral equations
Generalized Riesz potentials on spaces of homogeneous type
Dominating functions for the Navier–Stokes integral equations
A proof of Oseen's theorem through dominating functions
Functional spaces and multipliers

The differential and the integral Navier–Stokes equations

Uniform local estimates
Heat equation
Stokes equations
Oseen equations
Very weak solutions for the Navier–Stokes equations
Mild solutions for the Navier–Stokes equations
Suitable solutions for the Navier–Stokes equations

Mild solutions in Lebesgue or Sobolev spaces

Kato's mild solutions
Local solutions in the Hilbertian setting
Global solutions in the Hilbertian setting
Sobolev spaces
A commutator estimate
Lebesgue spaces
Maximal functions
Basic lemmas on real interpolation spaces
Uniqueness of L^3 solutions

Mild solutions in Besov or Morrey spaces

Morrey spaces
Morrey spaces and maximal functions
Uniqueness of Morrey solutions
Besov spaces
Regular Besov spaces
Triebel–Lizorkin spaces
Fourier transform and Navier–Stokes equations

The space BMO^-1 and the Koch and Tataru theorem

Koch and Tataru's theorem
Q-spaces
A special subclass of BMO^-1
Ill-posedness
Further results on ill-posedness
Large data for mild solutions
Stability of global solutions
Analyticity
Small data

Special examples of solutions

Symmetries for the Navier–Stokes equations
Two-and-a-half dimensional flows
Axisymmetrical solutions
Helical solutions
Brandolese's symmetrical solutions
Self-similar solutions
Stationary solutions
Landau's solutions of the Navier–Stokes equations
Time-periodic solutions
Beltrami flows

Blow up?

First criteria
Blow up for the cheap Navier–Stokes equation
Serrin's criterion
Some further generalizations of Serrin's criterion
Vorticity
Squirts

Leray's weak solutions

The Rellich lemma
Leray's weak solutions
Weak-strong uniqueness: the Prodi–Serrin criterion
Weak-strong uniqueness and Morrey spaces on the product space R × R^3
Almost strong solutions
Weak perturbations of mild solutions

Partial regularity results for weak solutions

Interior regularity
Serrin's theorem on interior regularity
O'Leary's theorem on interior regularity
Further results on parabolic Morrey spaces
Hausdorff measures
Singular times
The local energy inequality
The Caffarelli–Kohn–Nirenberg theorem on partial regularity
Proof of the Caffarelli–Kohn–Nirenberg criterion
Parabolic Hausdorff dimension of the set of singular points
On the role of the pressure in the Caffarelli, Kohn, and Nirenberg regularity theorem

A theory of uniformly locally L^2 solutions

Uniformly locally square integrable solutions
Local inequalities for local Leray solutions
The Caffarelli, Kohn, and Nirenberg ε-regularity criterion
A weak-strong uniqueness result

The L^3 theory of suitable solutions

Local Leray solutions with an initial value in L^3
Critical elements for the blow up of the Cauchy problem in L^3
Backward uniqueness for local Leray solutions
Seregin's theorem
Known results on the Cauchy problem for the Navier–Stokes equations in presence of a force
Local estimates for suitable solutions
Uniqueness for suitable solutions
A quantitative one-scale estimate for the Caffarelli–Kohn–Nirenberg regularity criterion
The topological structure of the set of suitable solutions
Escauriaza, Seregin, and Šverák's theorem

Self-similarity and the Leray–Schauder principle

The Leray–Schauder principle
Steady-state solutions
Self-similarity
Statement of Jia and Šverák's theorem
The case of locally bounded initial data
The case of rough data
Non-existence of backward self-similar solutions

α-models

Global existence, uniqueness and convergence issues for approximated equations
Leray's mollification and the Leray-α model
The Navier–Stokes α -model
The Clark- α model
The simplified Bardina model
Reynolds tensor

Other approximations of the Navier–Stokes equations

Faedo–Galerkin approximations
Frequency cut-off
Hyperviscosity
Ladyzhenskaya's model
Damped Navier–Stokes equations

Table of Contents

The Navier-Stokes Problem in the 21st Century : Table of Contents

Presentation of the Clay Millennium Prizes

The physical meaning of the Navier–Stokes equations

History of the equation

Classical solutions

A capacitary approach of the Navier–Stokes integral equations

The differential and the integral Navier–Stokes equations

Mild solutions in Lebesgue or Sobolev spaces

Mild solutions in Besov or Morrey spaces

The space BMO^-1 and the Koch and Tataru theorem

Special examples of solutions

Blow up?

Leray's weak solutions

Partial regularity results for weak solutions

A theory of uniformly locally L^2 solutions

The L^3 theory of suitable solutions

Self-similarity and the Leray–Schauder principle

α-models

Other approximations of the Navier–Stokes equations

Artificial compressibility

Conclusion

Notations and glossary

Bibliography

Index