An Introduction to Navier-Stokes Equation and Oceanography

1. Front Matter

   Pages i-xxvii

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2. Basic physical laws and units

   Pages 1-5

3. Radiation balance of atmosphere

   Pages 7-10

4. Conservations in ocean and atmosphere

   Pages 11-14

5. Sobolev spaces I

   Pages 15-22

6. Particles and continuum mechanics

   Pages 23-30

7. Conservation of mass and momentum

   Pages 31-36

8. Conservation of energy

   Pages 37-41

9. One-dimensional wave equation

   Pages 43-48

10. Nonlinear effects, shocks

    Pages 49-56

11. Sobolev spaces II

    Pages 57-62

12. Linearized elasticity

    Pages 63-67

13. Ellipticity conditions

    Pages 69-72

14. Sobolev spaces III

    Pages 73-76

15. Sobolev spaces IV

    Pages 77-81

16. Sobolev spaces V

    Pages 83-86

17. Sobolev embedding theorem

    Pages 87-93

18. Fixed point theorems

    Pages 95-100

19. Brouwer's topological degree

    Pages 101-105

20. Time-dependent solutions I

    Pages 107-112

In the spring of 1999, I taught (at CARNEGIEMELLON University) a graduate course entitled Partial Di?erential Equations Models in Oceanography, and I wrote lecture notes which I distributed to the students; these notes were then made available on the Internet, and they were distributed to the participants of a Summer School held in Lisbon, Portugal, in July 1999. After a few years, I feel it will be useful to make the text available to a larger audience by publishing a revised version. To an uninformed observer, it may seem that there is more interest in the Navier–Stokes equation nowadays, but many who claim to be interested show such a lack of knowledge about continuum mechanics that one may wonder about such a super?cial attraction. Could one of the Clay Millennium Prizes bethereasonbehindthisrenewedinterest?Readingthetextoftheconjectures to be solved for winning that particular prize leaves the impression that the subject was not chosen by people interested in continuum mechanics, as the selected questions have almost no physical content. Invariance by translation or scaling is mentioned, but why is invariance by rotations not pointed out 1 andwhyisGalileaninvariance omitted,asitistheessentialfactwhichmakes 1 Velocities involved for ordinary ?uids being much smaller than the velocity of light c, no relativistic corrections are necessary and Galilean invariance should then be used, but one should be aware that once the mathematical equation has been written it is not automatic that its solutions will only use velocities bounded by c.

• Navier-Stokes
• Navier-Stokes equation
• Sobolev space
• continuum mechanics
• mathematical tools
• partial differential equation
• partial differential equations
• wave equation

From the reviews:

"The book has its origin in a graduate course entitled ‘Partial Differential Equations Models in Oceanography’ presented by the author at Carnegie Mellon University in 1999. … The main objective is to teach readers to have a critical point of view concerning the partial differential equations of continuum mechanics and to show the need for developing new adapted mathematical tools. … Most of the theorems and lemmas are provided in the book or a corresponding reference is given. The bibliography contains 23 items." (Jürgen Socolowsky, Mathematical Reviews, Issue 2007 h)

"The book is written by a leading expert in the field and it will certainly be a valuable enhancement to the existing literature. This is a fascinating book consisting of 42 lectures which review some classical and modern aspects of Navier-Stokes Equations (NSE). … well organized and written in a lively and provoking style. … can be recommended to applied mathematicians and theoretical geophysicists working or interested in the field as well as being an appropriate material for graduate and postgraduate courses on the subject." (Andrzej Icha, Pure and Applied Geophysics, Vol. 165, 2008)

“The book consists of 44 lectures, completed with preface, introduction, detailed description of the lectures, bibliographical information, abbreviations and mathematical notation, references, and index. … this book is … a very good exposition of the topic it is dealing with. … The course had been intended for mathematicians in the first place, in the present book form, however, it will be a welcome reading, in its larger part, also for hydrodynamicists and other researchers in the field with less specialization in functional analysis.” (Tomislav Zlatanovski, Zentralblatt MATH, Vol. 1194, 2010)

• Department of Mathematical Sciences, Carnegie Mellon University, PA 15213-3890, Pittsburgh, USA

  Luc Tartar

Luc Tartar studied at Ecole Polytechnique in Paris, France, 1965-1967, where he was taught by Laurent Schwartz and Jacques-Louis Lions in mathematics, and by Jean Mandel in continuum mechanics.

He did research at Centre National de la Recherche Scientifique, Paris, France, 1968-1971, working under the direction of Jacques-Louis Lions for his thèse d'état, 1971.

He taught at Université Paris IX-Dauphine, Paris, France, 1971-1974, at University of Wisconsin, Madison, WI, 1974-1975, at Université de Paris-Sud, Orsay, France, 1975-1982.

He did research at Commissariat à l'Energie Atomique, Limeil, France, 1982-1987.

In 1987, he was elected Correspondant de l'Académie des Sciences, Paris, in the section Mécanique.

Since 1987 he has been teaching at Carnegie Mellon University, Pittsburgh, PA, where he has been University Professor of Mathematics since 1994.

Partly in collaboration with François Murat, he has specialized in the development of new mathematical tools for solving the partial differential equations of continuum mechanics (homogenization, compensated compactness, H-measures), pioneering the study of microstructures compatible with the partial differential equations describing the physical balance laws, and the constitutive relations.

He likes to point out the defects of many of the models which are used, as a natural way to achieve the goal of improving our understanding of mathematics and of continuum mechanics.

Policies and ethics