Includes bibliographical references and index.

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 NavierStokes equations--Boundaries and the NavierStokes Clay Millennium Problem -- The physical meaning of the NavierStokes equations--Frames of references --The convection theorem--Conservation of mass --Newton's second law --Pressure--Strain--Stress--The equations of hydrodynamics --The NavierStokes 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 NavierStokes 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 NavierStokes integral equations--The integral NavierStokes 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 NavierStokes integral equations--A proof of Oseen's theorem through dominating functions--Functional spaces and multipliers -- The differential and the integral NavierStokes equations -- Uniform local estimates --Heat equation --Stokes equations --Oseen equations --Very weak solutions for the NavierStokes equations --Mild solutions for the NavierStokes equations --Suitable solutions for the NavierStokes 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 L3 solutions -- Mild solutions in Besov or Morrey spaces -- Morrey spaces --Morrey spaces and maximal functions--Uniqueness of Morrey solutions--Besov spaces --Regular Besov spaces --TriebelLizorkin spaces --Fourier transform and NavierStokes 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 NavierStokes 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 NavierStokes equations--Time-periodic solutions --Beltrami flows -- Blow up? --First criteria--Blow up for the cheap NavierStokes 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 ProdiSerrin criterion--Weak-strong uniqueness and Morrey spaces on the product space R R3--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 CaffarelliKohnNirenberg theorem on partial regularity--Proof of the CaffarelliKohnNirenberg 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 L2 solutions--Uniformly locally square integrable solutions--Local inequalities for local Leray solutions--The Caffarelli, Kohn, and Nirenberg -regularity criterion--A weak-strong uniqueness result -- The L3 theory of suitable solutions--Local Leray solutions with an initial value in L3--Critical elements for the blow up of the Cauchy problem in L3--Backward uniqueness for local Leray solutions--Seregin's theorem--Known results on the Cauchy problem for the NavierStokes equations in presence of a force--Local estimates for suitable solutions--Uniqueness for suitable solutions--A quantitative one-scale estimate for the CaffarelliKohnNirenberg regularity criterion--The topological structure of the set of suitable solutions--Escauriaza, Seregin, and verk's theorem -- Self-similarity and the LeraySchauder principle--The LeraySchauder principle--Steady-state solutions--Self-similarity--Statement of Jia and verk's theorem--The case of locally bounded initial data--The case of rough data--Non-existence of backward self-similar solutions -- ---models--Global existence, uniqueness and convergence issues for approximated equations --Leray's mollification and the Leray- model--The NavierStokes -model--The Clark- model--The simplified Bardina model--Reynolds tensor -- Other approximations of the NavierStokes equations--FaedoGalerkin approximations --Frequency cut-off --Hyperviscosity --Ladyzhenskaya's model--Damped NavierStokes 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.

Up-to-Date Coverage of the Navier-Stokes Equation from an Expert in Harmonic Analysis The complete resolution of the Navier-Stokes equation-one of the Clay Millennium Prize Problems-remains an important open challenge in partial differential equations (PDEs) research despite substantial studies on turbulence and three-dimensional fluids. The Navier-Stokes Problem in the 21st Century provides a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics. The book focuses on incompressible deterministic Navier-Stokes equations in the case of a fluid filling the whole space. It explores the meaning of the equations, open problems, and recent progress. It includes classical results on local existence and studies criterion for regularity or uniqueness of solutions. The book also incorporates historical references to the (pre)history of the equations as well as recent references that highlight active mathematical research in the field.

Also available in print format.