Navier-Stokes equations in planar domainsMatania Ben-Artzi, Jean-Pierre Croisille, Dalia Fishelov.

Includes bibliographies and index.

Pt. I. Basic theory. 1. Introduction. 1.1. Functional notation -- 2. Existence and uniqueness of smooth solutions. 2.1. The linear convection-diffusion equation. 2.2. Proof of theorem 2.1. 2.3. Existence and uniqueness in Hölder spaces. 2.4. Notes for chapter 2 -- 3. Estimates for smooth solutions. 3.1. Estimates involving [symbol. 3.2. Estimates involving [symbol. 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 [symbol. 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. 6.3. Stability of steady states. 6.4. Notes for chapter 6 -- A. Some theorems from functional analysis. A.1. The Calderón-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 [symbol] and the heat kernel -- part 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. 9.3. Matrix representation of the three-point Laplacian. 9.4. Notes for chapter 9 -- 10. From Hermitian derivative to the compact discrete biharmonic operator. 10.1. 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 rectangle. 11.2. The biharmonic problem in an irregular domain. 11.3. Notes for chapter 11 -- 12. Compact approximation of the Navier-Stokes equations in streamfunction formulation. 12.1. The Navier-Stokes equations in streamfunction formulation. 12.2. Discretizing the streamfunction equation. 12.3. Convergence of the scheme. 12.4. Notes for chapter 12 -- B. Eigenfunction approach for [symbol. B.1. Some basic properties of the equation. B.2. The discrete approximation -- 13. Fully discrete approximation of the Navier-Stokes equations. 13.1. Fourth-order approximation in space. 13.2. A time-stepping discrete scheme. 13.3. Numerical results. 13.4. Notes for chapter 13 -- 14. Numerical simulations of the driven cavity problem. 14.1. Second-order scheme for the driven cavity problem. 14.2. Fourth-order scheme for the driven cavity problem. 14.3. Double-driven cavity problem. 14.4. Notes for chapter 14.

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 problems such as "driven cavity" and "double-driven cavity". A comprehensive treatment of the mathematical theory developed in the last 15 years is elaborated, heretofore never presented in other books. It gives a detailed account of the modern compact schemes based on a "pure streamfunction" approach. In particular, a complete proof of convergence is given for the full nonlinear problem. This volume aims to present a variety of numerical test problems. It is therefore well positioned as a reference for both theoretical and applied mathematicians, as well as a text that can be used by graduate students pursuing studies in (pure or applied) mathematics, fluid dynamics and mathematical physics.

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