Mathematical theory of incompressible nonviscous fluids
著者
書誌事項
Mathematical theory of incompressible nonviscous fluids
（Applied mathematical sciences, v. 96）
Springer-Verlag, c1994
- : us
- : gw
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注記
Includes bibliographical references (p. 272-280) and index
内容説明・目次
- 巻冊次
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: us ISBN 9780387940441
内容説明
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 is played by incompressible nonviscous (sometimes called perfect) flows. This is a mathematical model consisting essentially of an evolution equation (the Euler equation) for the velocity field of fluids. Such an equation, which is nothing other than the Newton laws plus some additional structural hypo theses, was discovered by Euler in 1755, and although it is more than two centuries old, many fundamental questions concerning its solutions are still open. In particular, it is not known whether the solutions, for reasonably general initial conditions, develop singularities in a finite time, and very little is known about the long-term behavior of smooth solutions. These and other basic problems are still open, and this is one of the reasons why the mathe matical theory of perfect flows is far from being completed. Incompressible flows have been attached, by many distinguished mathe maticians, with a large variety of mathematical techniques so that, today, this field constitutes a very rich and stimulating part of applied mathematics.
目次
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.
- 巻冊次
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: gw ISBN 9783540940449
内容説明
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 set-up, 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 begin 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.
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