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Table of Contents

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

Artificial compressibility

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

Conclusion

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

Notations and glossary

Bibliography

Index

members/lemarie/ns21toc.txt · Last modified: 2016/03/23 19:30 by Pierre Gilles Lemarié

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