• 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 L^3 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