The NavierStokes problem in the 21st century
 Responsibility
 Pierre Gilles LemariéRieusset (Université d'Évry Val d'Essonne, Evry, France).
 Publication
 Boca Raton, FL : CRC Press, Taylor & Francis Group, [2016]
 Copyright notice
 ©2016
 Physical description
 xxii, 718 pages ; 25 cm
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA901 .L46 2016  Unknown 
More options
Description
Creators/Contributors
 Author/Creator
 Lemarié, Pierre Gilles, 1960 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 677708) and index.
 Contents

 Presentation of the Clay Millennium Prizes Regularity of the threedimensional 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, SaintVenant, 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 BMO1 and the Koch and Tataru theorem Koch and Tataru's theorem Qspaces A special subclass of BMO1 Illposedness Further results on illposedness Large data for mild solutions Stability of global solutions Analyticity Small data
 Special examples of solutions Symmetries for the NavierStokes equations Twoandahalf dimensional flows Axisymmetrical solutions Helical solutions Brandolese's symmetrical solutions Selfsimilar solutions Stationary solutions Landau's solutions of the NavierStokes equations Timeperiodic 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 Weakstrong uniqueness: the ProdiSerrin criterion Weakstrong uniqueness and Morrey spaces on the product space R x 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 epsilonregularity criterion A weakstrong 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 onescale estimate for the CaffarelliKohnNirenberg regularity criterion The topological structure of the set of suitable solutions Escauriaza, Seregin, and Sverak's theorem
 Selfsimilarity and the LeraySchauder principle The LeraySchauder principle Steadystate solutions Selfsimilarity Statement of Jia and Sverak's theorem The case of locally bounded initial data The case of rough data Nonexistence of backward selfsimilar solutions
 alphamodels Global existence, uniqueness and convergence issues for approximated equations Leray's mollification and the Lerayalpha model The NavierStokes alpha model The Clark alpha model The simplified Bardina model Reynolds tensor
 Other approximations of the NavierStokes equations FaedoGalerkin approximations Frequency cutoff 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.
 (source: Nielsen Book Data)
 Summary

UptoDate Coverage of the NavierStokes Equation from an Expert in Harmonic Analysis The complete resolution of the NavierStokes equationone of the Clay Millennium Prize Problemsremains an important open challenge in partial differential equations (PDEs) research despite substantial studies on turbulence and threedimensional fluids. The NavierStokes Problem in the 21st Century provides a selfcontained guide to the role of harmonic analysis in the PDEs of fluid mechanics. The book focuses on incompressible deterministic NavierStokes 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2016
 Copyright date
 2016
 Note
 "A Chapman & Hall book."
 ISBN
 9781466566217 (alk. paper)
 1466566213 (alk. paper)