Author | Marchioro, Carlo. author |
---|---|
Title | Mathematical Theory of Incompressible Nonviscous Fluids [electronic resource] / by Carlo Marchioro, Mario Pulvirenti |
Imprint | New York, NY : Springer New York, 1994 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4284-0 |
Descript | XII, 284 p. online resource |
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 (dโAlembert 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