Carlo Marchioro, Mario Pulvirenti

Mathematical Theory of Incompressible Nonviscous Fluids

• Gebundenes Buch

Produktbeschreibung

This book deals with fluid dynamics of incompressible non-viscous fluids. The main goal is to present an argument of large interest for physics, and applications in a rigorous logical and mathematical setup, therefore avoiding cumbersome technicalities. Classical as well as modern mathematical developments are illustrated in this book, which should fill a gap in the present literature. The book does not require a deep mathematical knowledge. The required background is a good understanding of classical arguments of mathematical analysis, including the basic elements of ordinary and partial differential equations, measure theory and analytic functions, and a few notions of potential theory and functional analysis. The contents of the book begins with the Euler equation, construction of solutions, stability of stationary solutions of the Euler equation. It continues with the vortex model, approximation methods, evolution of discontinuities, and concludes with turbulence.

Produktdetails

• Applied Mathematical Sciences
• Verlag: Springer, Berlin
• Seitenzahl: 283
• Englisch
• Abmessung: 245mm
• Gewicht: 568g
• ISBN-13: 9783540940449
• Artikelnr.: 26008293

Inhaltsangabe

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. SpatiallyInhomogeneous 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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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. SpatiallyInhomogeneous 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.