====== 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 =====