The Three-Dimensional Navier–Stokes Equations: Classical Theory: 157 (Cambridge Studies in Advanced Mathematics, Series Number 157)
Cambridge University Press, 9/7/2016
EAN 9781107019669, ISBN10: 1107019664
Hardcover, 484 pages, 25.4 x 17.8 x 3.2 cm
Language: English
Originally published in English
A rigorous but accessible introduction to the mathematical theory of the three-dimensional Navier–Stokes equations, this book provides self-contained proofs of some of the most significant results in the area, many of which can only be found in research papers. Highlights include the existence of global-in-time Leray–Hopf weak solutions and the local existence of strong solutions; the conditional local regularity results of Serrin and others; and the partial regularity results of Caffarelli, Kohn, and Nirenberg. Appendices provide background material and proofs of some 'standard results' that are hard to find in the literature. A substantial number of exercises are included, with full solutions given at the end of the book. As the only introductory text on the topic to treat all of the mainstream results in detail, this book is an ideal text for a graduate course of one or two semesters. It is also a useful resource for anyone working in mathematical fluid dynamics.
Part I. Weak and Strong Solutions
1. Function spaces
2. The Helmholtz–Weyl decomposition
3. Weak formulation
4. Existence of weak solutions
5. The pressure
6. Existence of strong solutions
7. Regularity of strong solutions
8. Epochs of regularity and Serrin's condition
9. Robustness of regularity
10. Local existence and uniqueness in H1/2
11. Local existence and uniqueness in L3
Part II. Local and Partial Regularity
12. Vorticity
13. The Serrin condition for local regularity
14. The local energy inequality
15. Partial regularity I – dimB(S) ≤ 5/3
16. Partial regularity II – dimH(S) ≤ 1
17. Lagrangian trajectories
A. Functional analysis
miscellaneous results
B. Calderón–Zygmund Theory
C. Elliptic equations
D. Estimates for the heat equation
E. A measurable-selection theorem
Solutions to exercises
References
Index.