Mathematical Theory of Incompressible Nonviscous Fluids

Mathematical Theory of Incompressible Nonviscous Fluids by Carlo Marchioro, Mario Pulvirenti.

Fluid dynamics is an ancient science incredibly alive today. Modern technol­ ogy and new needs require a deeper knowledge of the behavior of real fluids, and new discoveries or steps forward pose, quite often, challenging and diffi­ cult new mathematical {::oblems. In this framework, a special role...

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Bibliographic Details
Main Authors: Marchioro, Carlo (Author), Pulvirenti, Mario (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 1994.
Edition:1st ed. 1994.
Series:Applied Mathematical Sciences, 96
Springer eBook Collection.
Subjects:
Mathematical analysis.
Analysis (Mathematics).
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 General Considerations on the Euler Equation
  • 1.1. The Equation of Motion of an Ideal Incompressible Fluid
  • 1.2. Vorticity and Stream Function
  • 1.3. Conservation Laws
  • 1.4. Potential and Irrotational Flows
  • 1.5. Comments
  • Appendix 1.1 (Liouville Theorem)
  • Appendix 1.2 (A Decomposition Theorem)
  • Appendix 1.3 (Kutta-Joukowski Theorem and Complex Potentials)
  • Appendix 1.4 (d’Alembert Paradox)
  • Exercises
  • 2 Construction of the Solutions
  • 2.1. General Considerations
  • 2.2. Lagrangian Representation of the Vorticity
  • 2.3. Global Existence and Uniqueness in Two Dimensions
  • 2.4. Regularity Properties and Classical Solutions
  • 2.5. Local Existence and Uniqueness in Three Dimensions
  • 2.6. Some Heuristic Considerations on the Three-Dimensional Motion
  • 2.7. Comments
  • Appendix 2.1 (Integral Inequalities)
  • Appendix 2.2 (Some Useful Inequalities)
  • Appendix 2.3 (Quasi-Lipschitz Estimate)
  • Appendix 2.4 (Regularity Estimates)
  • Exercises
  • 3 Stability of Stationary Solutions of the Euler Equation
  • 3.1. A Short Review of the Stability Concept
  • 3.2. Sufficient Conditions for the Stability of Stationary Solutions: The Arnold Theorems
  • 3.3. Stability in the Presence of Symmetries
  • 3.4. Instability
  • 3.5. Comments
  • Exercises
  • 4 The Vortex Model
  • 4.1. Heuristic Introduction
  • 4.2. Motion of Vortices in the Plane
  • 4.3. The Vortex Motion in the Presence of Boundaries
  • 4.4. A Rigorous Derivation of the Vortex Model
  • 4.5. Three-Dimensional Models
  • 4.6. Comments
  • Exercises
  • 5 Approximation Methods
  • 5.1. Introduction
  • 5.2. Spectral Methods
  • 5.3. Vortex Methods
  • 5.4. Comments
  • Appendix 5.1 (On K-R Distance)
  • Exercises
  • 6 Evolution of Discontinuities
  • 6.1. Vortex Sheet
  • 6.2. Existence and Behavior of the Solutions
  • 6.3. Comments
  • 6.4. Spatially Inhomogeneous Fluids
  • 6.5. Water Waves
  • 6.6. Approximations
  • Appendix 6.1 (Proof of a Theorem of the Cauchy-Kowalevski Type)
  • Appendix 6.2 (On Surface Tension)
  • 7 Turbulence
  • 7.1. Introduction
  • 7.2. The Onset of Turbulence
  • 7.3. Phenomenological Theories
  • 7.4. Statistical Solutions and Invariant Measures
  • 7.5. Statistical Mechanics of Vortex Systems
  • 7.6. Three-Dimensional Models for Turbulence
  • References.

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