Navier-Stokes Equations in Planar Domains.
This volume deals with the classical Navier-Stokes system of equations governing the planar flow of incompressible, viscid fluid. It is a first-of-its-kind book, devoted to all aspects of the study of such flows, ranging from theoretical to numerical, including detailed accounts of classical test pr...
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| Format: | eBook |
| Language: | English |
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Singapore :
World Scientific Publishing Company,
2013.
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Table of Contents:
- Preface; Contents; Part I Basic Theory; 1. Introduction; 1.1 Functional notation; 2. Existence and Uniqueness of Smooth Solutions; 2.1 The linear convection-diffusion equation; 2.1.1 Unbounded coefficient at t = 0; 2.2 Proof of Theorem 2.1; 2.3 Existence and uniqueness in Holder spaces; 2.4 Notes for Chapter 2; 3. Estimates for Smooth Solutions; 3.1 Estimates involving 0 L1(R2); 3.1.1 Refinement for short time; 3.2 Estimates involving 0 Lp(R2); 3.3 Estimating derivatives; 3.4 Notes for Chapter 3; 4. Extension of the Solution Operator; 4.1 An intermediate extension.
- 4.2 Extension to initial vorticity in L1(R2)4.3 Notes for Chapter 4; 5. Measures as Initial Data; 5.1 Uniqueness for general initial measures; 5.2 Notes for Chapter 5; 6. Asymptotic Behavior for Large Time; 6.1 Decay estimates for large time; 6.2 Initial data with stronger spatial decay; 6.2.1 Scaling variables and invariant manifolds; 6.3 Stability of steady states; 6.4 Notes for Chapter 6; A. Some Theorems from Functional Analysis; A.1 The Calder ́on-Zygmund Theorem; A.2 Young's and the Hardy-Littlewood-Sobolev Inequalities; A.3 The Riesz-Thorin Interpolation Theorem.
- A.4 Finite Borel measures in R2 and the heat kernelPart II Approximate Solutions; 7. Introduction; 8. Notation; 8.1 One-dimensional discrete setting; 8.2 Two-dimensional discrete setting; 9. Finite Difference Approximation to Second-Order Boundary-Value Problems; 9.1 The principle of finite difference schemes; 9.2 The three-point Laplacian; 9.2.1 General setting; 9.2.2 Maximum principle analysis; 9.2.3 Coercivity and energy estimate; 9.3 Matrix representation of the three-point Laplacian; 9.3.1 Continuous and discrete eigenfunctions; 9.3.2 Convergence analysis; 9.4 Notes for Chapter 9.
- 10. From Hermitian Derivative to the Compact Discrete Biharmonic Operator10.1 The Hermitian derivative operator; 10.2 A finite element approach to the Hermitian derivative; 10.3 The three-point biharmonic operator; 10.4 Accuracy of the three-point biharmonic operator; 10.5 Coercivity and stability properties of the three-point biharmonic operator; 10.6 Matrix representation of the three-point biharmonic operator; 10.7 Convergence analysis using the matrix representation; 10.8 Notes for Chapter 10; 11. Polynomial Approach to the Discrete Biharmonic Operator.
- 11.1 The biharmonic problem in a rectangle11.1.1 General setting; 11.1.2 The nine-point compact biharmonic operator; 11.1.3 Accuracy of the discrete biharmonic operator; 11.1.4 Coercivity and convergence properties of the discrete biharmonic operator; 11.2 The biharmonic problem in an irregular domain; 11.2.1 Embedding a Cartesian grid in an irregular domain; 11.2.2 The biharmonic 2 operator; 11.2.2.1 Approximating the data using a sixth-order polynomial; 11.2.2.2 Choice of; 11.2.3 Accuracy of the finite difference biharmonic operator; 11.2.3.1 Proof of Step I of Theorem 11.18.