Mathematical Theory of Incompressible Nonviscous Fluids

Carlo Marchioro, Mario Pulvirenti (Autoren)

Buch | Hardcover

284 Seiten

1993
Springer-Verlag New York Inc.
978-0-387-94044-1 (ISBN)

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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.

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.

Erscheint lt. Verlag	5.11.1993
Reihe/Serie	Applied Mathematical Sciences ; 96
Zusatzinfo	XII, 284 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	156 x 234 mm
Themenwelt	Schulbuch / Wörterbuch
	Mathematik / Informatik ► Mathematik ► Analysis
	Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
ISBN-10	0-387-94044-8 / 0387940448
ISBN-13	978-0-387-94044-1 / 9780387940441
Zustand	Neuware