The Three Dimensional Navier Stokes Equations Classical Theory 1st Edition by James Robinson, José Rodrigo, Witold Sadowski ISBN 978-1107019669 1107019669

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Product details: 

ISBN 10: 1107019669

ISBN 13: 978-1107019669

Author:  James C. Robinson, José L. Rodrigo, Witold Sadowski 

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.

Table of contents: 

1. Function Spaces
1.1 Domain of the Flow
1.2 Derivatives
1.3 Spaces of Continuous and Differentiable Functions
1.4 Lebesgue Spaces
1.5 Fourier Expansions
1.6 Sobolev Spaces Wk,p
1.7 Sobolev Spaces Hs with s≥ 0
1.8 Dual Spaces
1.9 Bochner Spaces

2. The Helmholtz-Weyl Decomposition
2.1 The Helmholtz-Weyl Decomposition on the Torus
2.2 The Helmholtz-Weyl Decomposition in ℝ³
2.3 The Stokes Operator
2.4 The Helmholtz-Weyl Decomposition of L

3. Weak Formulation
3.1 Weak Formulation
3.2 Basic Properties of Weak Solutions
3.3 Alternative Spaces of Test Functions
3.4 Equivalent Weak Formulation
3.5 Uniqueness of Weak Solutions in Dimension Two

4. Existence of Weak Solutions
4.1 The Galerkin Method
4.2 Existence of Weak Solutions on Bounded Domains
4.3 The Strong Energy Inequality
4.4 Existence of Weak Solutions on the Whole Space
4.5 The Aubin-Lions Lemma

5. The Pressure
5.1 Solving for the Pressure on T³ and ℝ³
5.2 Distributional Solutions in the Absence of Boundaries
5.3 Additional Estimates on Weak Solutions
5.4 Pressure in a Bounded Domain
5.5 Applications of Pressure Estimates

6. Existence of Strong Solutions
6.1 General Properties of Strong Solutions
6.2 Local Existence of Strong Solutions
6.3 Weak-Strong Uniqueness and Blowup
6.4 Global Existence for Small Data in V
6.5 Global Strong Solutions in the Two-Dimensional Case
6.6 Strong Solutions on the Whole Space

7. Regularity of Strong Solutions
7.1 Regularity in Space
7.2 Regularity in Space-Time

8. Epochs of Regularity and Serrin’s Condition
8.1 The Putative Set of Singular Times
8.2 The Box-Counting and Hausdorff Dimensions
8.3 Epochs of Regularity
8.4 More Bounds on Weak Solutions
8.5 Serrin’s Condition
8.6 Epochs of Regularity on the Whole Space

9. Robustness of Regularity and Convergence of Galerkin Approximations
9.1 Robustness of Strong Solutions
9.2 Convergence of Galerkin Approximations

10. Local Existence and Uniqueness in H¹/₂
10.1 Critical Spaces
10.2 Fractional Sobolev Spaces and Criticality of Ĥ¹/₂
10.3 Local Existence for Initial Data in A¹/₂
10.4 An Auxiliary ODE Lemma

11. Local Existence and Uniqueness in L³
11.1 Preliminaries
11.2 Local Existence in L³
11.3 A Proof of Lemma 11.2 on T³

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Tags: James Robinson, José Rodrigo, Witold Sadowski, The Three Dimensional, Stokes Equations