• 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

• 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

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

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

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

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

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 L3 solutions

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

• 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

• 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

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

• 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

• 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

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

• 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

• 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

• 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

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

• Temam's model
• Vishik and Fursikov's model
• Hyperbolic approximation

• Energy inequalities
• Critical spaces for mild solutions
• Models for the (potential) blow up
• The method of critical elements