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Lemarie-Rieusset

The Navier-Stokes Problem in the 21st Century

Medium: Buch
ISBN: 978-0-367-48726-3
Verlag: Chapman and Hall/CRC
Erscheinungstermin: 08.12.2023
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Praise for the first edition “The author is an outstanding expert in harmonic analysis who has made important contributions. The book contains rigorous proofs of a number of the latest results in the field. I strongly recommend the book to postgraduate students and researchers working on challenging problems of harmonic analysis and mathematical theory of Navier-Stokes equations."—Gregory Seregin, St Hildas College, Oxford University

“"This is a great book on the mathematical aspects of the fundamental equations of hydrodynamics, the incompressible Navier-Stokes equations. It covers many important topics and recent results and gives the reader a very good idea about where the theory stands at present.”—Vladimir Sverak, University of Minnesota

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, Second Edition continues to provide a self-contained guide to the role of harmonic analysis in the PDEs of fluid mechanics, now revised to include fresh examples, theorems, results, and references that have become relevant since the first edition published in 2016.

Produkteigenschaften

  • Artikelnummer: 9780367487263
  • Medium: Buch
  • ISBN: 978-0-367-48726-3
  • Verlag: Chapman and Hall/CRC
  • Erscheinungstermin: 08.12.2023
  • Sprache(n): Englisch
  • Auflage: 2. Auflage 2023
  • Produktform: Gebunden
  • Gewicht: 1631 g
  • Seiten: 778
  • Format (B x H x T): 183 x 260 x 46 mm
  • Ausgabetyp: Kein, Unbekannt
  • Vorauflage: 978-1-4665-6621-7

Themen

Autoren/Hrsg.

Autoren

1. Presentation of the Clay Millennium Prizes. 1.1. Regularity of the three-dimensional fluid flows: a mathematical challenge for the 21st century. 1.2. The Clay Millennium Prizes. 1.3. The Clay Millennium Prize for the Navier-Stokes equations. 1.4. Boundaries and the Navier-Stokes Clay Millennium Problem. 2. The physical meaning of the Navier-Stokes equations. 2.1. Frames of references. 2.2. The convection theorem. 2.3. Conservation of mass. 2.4. Newton's second law. 2.5. Pressure. 2.6. Strain. 2.7. Stress. 2.8. The equations of hydrodynamics. 2.9. The Navier-Stokes equations. 2.10. Vorticity. 2.11. Boundary terms. 2.12. Blow-up. 2.13. Turbulence. 3. History of the equation. 3.1. Mechanics in the Scientific Revolution era. 3.2. Bernoulli's Hydrodymica. 3.3. D'Alembert. 3.4. Euler. 3.5. Laplacian physics. 3.6. Navier, Cauchy, Poisson, Saint-Venant, and Stokes. 3.7. Reynolds. 3.8. Oseen, Leray, Hopf, and Ladyzhenskaya. 3.9. Turbulence models. 4. Classical solutions. 4.1. The heat kernel. 4.2. The Poisson equation. 4.3. The Helmholtz decomposition. 4.4. The Stokes equation. 4.5. The Oseen tensor. 4.6. Classical solutions for the Navier-Stokes problem. 4.7. Maximal classical solutions and estimates in L8 norms. 4.8. Small data. 4.9. Spatial asymptotics. 4.10. Spatial asymptotics for the vorticity. 4.11. Maximal classical solutions and estimates in L2 norms. 4.12. Intermediate conclusion. 5. A capacitary approach of the Navier-Stokes integral equations. 5.1. The integral Navier-Stokes problem. 5.2. Quadratic equations in Banach spaces. 5.3. A capacitary approach of quadratic integral equations. 5.4. Generalized Riesz potentials on spaces of homogeneous type. 5.5. Dominating functions for the Navier-Stokes integral equations. 5.6. Oseen's theorem and dominating functions. 5.7. Functional spaces and multipliers. 6. The differential and the integral Navier-Stokes equations. 6.1. Very weak solutions for the Navier-Stokes equations. 6.2. Heat equation. 6.3. The Leray projection operator. 6.4. Stokes equations. 6.5. Oseen equations. 6.6. Mild solutions for the Navier-Stokes equations. 6.7. Suitable solutions for the Navier-Stokes equations. 7. Mild solutions in Lebesgue or Sobolev spaces. 7.1. Kato's mild solutions. 7.2. Local solutions in the Hilbertian setting. 7.3. Global solutions in the Hilbertian setting. 7.4. Sobolev spaces. 7.5. A commutator estimate. 7.6. Lebesgue spaces. 7.7. Maximal functions. 7.8. Basic lemmas on real interpolation spaces. 7.9. Uniqueness of L3 solutions. 8. Mild solutions in Besov or Morrey spaces. 8.1. Morrey spaces. 8.2. Morrey spaces and maximal functions. 8.3. Uniqueness of Morrey solutions. 8.4. Besov spaces. 8.5. Regular Besov spaces. 8.6. Triebel-Lizorkin spaces. 8.7. Fourier transform and Navier-Stokes equations. 8.8. The cheap Navier-Stokes equation. 8.9. Plane waves. 9. The space BMO-1 and the Koch and Tataru theorem. 9.1. The Koch and Tataru theorem. 9.2. A variation on the Koch and Tataru theorem. 9.3. Q-spaces. 9.4. A special subclass of BMO-1. 9.5. Ill-posedness. 9.6. Further results on ill-posedness. 9.7. Large data for mild solutions. 9.8. Stability of global solutions. 9.9. Analyticity. 9.10 Small data. 10. Special examples of solutions. 10.1 Symmetries for the Navier-Stokes equations. 10.2 Two-and-a-half dimensional flows. 10.3 Axisymmetrical solutions. 10.4 Helical solutions. 10.5 Brandolese's symmetrical solutions. 10.6 Self-similar solutions. 10.7 Stationary solutions. 10.8 Landau's solutions of the Navier-Stokes equations. 10.9 Time-periodic solutions. 10.10 Beltrami flows. 11. Blow-up? 11.1. First criteria. 11.2. Blow-up for the cheap Navier-Stokes equation. 11.3. Serrin's criterion. 11.4. A remark on Serrin's criterion and Leray's criterion. 11.5. Some further generalizations of Serrin's criterion. 11.6. Vorticity. 11.7. Squirts. 11.8. Eigenvalues of the strain matrix. 12. Leray's weak solutions. 12.1. The Rellich lemma.