This is the second of four volumes on the Navier-Stokes equations, specifically on Nonlinear Stationary Problems. The volumes deal with the fundamental mathematical properties of the Navier-Stokes equations, such as existence, regularity and uniqueness of solutions, and, for unbounded domains, their asymptotic behavior. The work is an up-to-date and detailed investigation of these problems for motions in domains of different types: bounded, exterior and domain with noncompact boundaries. Throughout the work, main problems which, so far, remain open are pointed out and for some of these…mehr
This is the second of four volumes on the Navier-Stokes equations, specifically on Nonlinear Stationary Problems. The volumes deal with the fundamental mathematical properties of the Navier-Stokes equations, such as existence, regularity and uniqueness of solutions, and, for unbounded domains, their asymptotic behavior. The work is an up-to-date and detailed investigation of these problems for motions in domains of different types: bounded, exterior and domain with noncompact boundaries. Throughout the work, main problems which, so far, remain open are pointed out and for some of these conjectures are offered. New results are presented throughout, while several classical subjects are treated in a completely original way. The work is mathematically self contained, requiring no specific background. The 200-plus exercises along with the chapter summaries and questions make this an excellent textbook for any theoretical Fluid Mechanics course; it is suitable as well for self teaching. It is set up to remain useful as a reference or dictionary.
VIII Steady Navier-Stokes Flow in Bounded Domains.- VIII. 1 Generalised Solutions. Preliminary Considerations.- VIII.2 On the Uniqueness of Generalised Solutions.- VIII.3 Existence and Uniqueness with Homogeneous Boundary Data.- VIII.4 Existence and Uniqueness with Nonhomogeneous Boundary Data.- VIII.5 Regularity of Generalised Solutions.- VIII.6 Limit of Infinite Viscosity: Transition to the Stokes Problem.- VIII. 7 Notes for the Chapter.- IX Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains.- IX.1 Generalised Solutions. Preliminary Considerations and Regularity Properties.- IX.2 On the Validity of the Energy Equation for Generalised Solutions.- IX.3 Some Uniqueness Results.- IX.4 Existence of Generalised Solutions.- IX.5 The Energy Equation and Uniqueness for Generalised Solutions when ?? ? 0.- IX.6 On the Asymptotic Behaviour of Generalised Solutions: Preliminary Results and Representation Formulas.- IX.7 Global Summability of Generalised Solutions when ?? ? 0.- IX.8 The Asymptotic Structure of Generalised Solutions when ?? ? 0.- IX.9 On the Asymptotic Structure of Generalised Solutions when ?? ? 0.- IX. 10 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem.- IX. 11 Notes for the Chapter.- X Steady Navier-Stokes Flow in Two-Dimensional Exterior Domains.- X.1 Generalised Solutions and D-Solutions.- X.2 On the Uniqueness of Generalised Solutions.- X.3 On the Asymptotic Behaviour of D-Solutions.- X.4 Existence and Uniqueness of Solutions when ?? ? 0.- X.5 Global Summability of Generalised Solutions when ?? ? 0.- X.6 The Asymptotic Structure of Generalised Solutions when ?? ? 0.- X.7 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem.- X.8 Notes for the Chapter.- XI Steady Navier-StokesFlow in Domains with Unbounded Boundaries.- XI.1 Leray's Problem: Generalised Solutions and Related Properties.- XI.2 On the Uniqueness of Generalised Solutions to Leray's Problem.- XI.3 Existence and Uniqueness of Solutions to Leray's Problem.- XI.4 Decay Estimates for Steady Flow in a Semi-Infinite Straight Channel.- XI.5 Flow in an Aperture Domain. Generalised Solutions and Related Properties.- XI.6 Energy Equation and Uniqueness for Flows in an Aperture Domain.- XI.7 Existence and Uniqueness of Flows in an Aperture Domain.- XI.8 Global Summability of Generalised Solutions for Flow in an Aperture Domain.- XI.9 Asymptotic Structure of Generalised Solutions for Flow in an Aperture Domain.- XI. 10 Notes for the Chapter.
VIII Steady Navier-Stokes Flow in Bounded Domains.- VIII. 1 Generalised Solutions. Preliminary Considerations.- VIII.2 On the Uniqueness of Generalised Solutions.- VIII.3 Existence and Uniqueness with Homogeneous Boundary Data.- VIII.4 Existence and Uniqueness with Nonhomogeneous Boundary Data.- VIII.5 Regularity of Generalised Solutions.- VIII.6 Limit of Infinite Viscosity: Transition to the Stokes Problem.- VIII. 7 Notes for the Chapter.- IX Steady Navier-Stokes Flow in Three-Dimensional Exterior Domains.- IX.1 Generalised Solutions. Preliminary Considerations and Regularity Properties.- IX.2 On the Validity of the Energy Equation for Generalised Solutions.- IX.3 Some Uniqueness Results.- IX.4 Existence of Generalised Solutions.- IX.5 The Energy Equation and Uniqueness for Generalised Solutions when ?? ? 0.- IX.6 On the Asymptotic Behaviour of Generalised Solutions: Preliminary Results and Representation Formulas.- IX.7 Global Summability of Generalised Solutions when ?? ? 0.- IX.8 The Asymptotic Structure of Generalised Solutions when ?? ? 0.- IX.9 On the Asymptotic Structure of Generalised Solutions when ?? ? 0.- IX. 10 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem.- IX. 11 Notes for the Chapter.- X Steady Navier-Stokes Flow in Two-Dimensional Exterior Domains.- X.1 Generalised Solutions and D-Solutions.- X.2 On the Uniqueness of Generalised Solutions.- X.3 On the Asymptotic Behaviour of D-Solutions.- X.4 Existence and Uniqueness of Solutions when ?? ? 0.- X.5 Global Summability of Generalised Solutions when ?? ? 0.- X.6 The Asymptotic Structure of Generalised Solutions when ?? ? 0.- X.7 Limit of Vanishing Reynolds Number: Transition to the Stokes Problem.- X.8 Notes for the Chapter.- XI Steady Navier-StokesFlow in Domains with Unbounded Boundaries.- XI.1 Leray's Problem: Generalised Solutions and Related Properties.- XI.2 On the Uniqueness of Generalised Solutions to Leray's Problem.- XI.3 Existence and Uniqueness of Solutions to Leray's Problem.- XI.4 Decay Estimates for Steady Flow in a Semi-Infinite Straight Channel.- XI.5 Flow in an Aperture Domain. Generalised Solutions and Related Properties.- XI.6 Energy Equation and Uniqueness for Flows in an Aperture Domain.- XI.7 Existence and Uniqueness of Flows in an Aperture Domain.- XI.8 Global Summability of Generalised Solutions for Flow in an Aperture Domain.- XI.9 Asymptotic Structure of Generalised Solutions for Flow in an Aperture Domain.- XI. 10 Notes for the Chapter.
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