Roma, 2000. 502 pThe imagination is stricken by the substantial conceptual identity between the problems met in the theoretical study of physical phenomena. It is absolutely unexpected and surprising, whether one studies equilibrium statistical mechanics, or quantum fleld theory, or solid state physics, or celestial mechanics, harmonic analysis, elasticity, general relativity or fluid mechanics and chaos in turbulence Indice Generalities on Continua Continua General and incompressible equations The rescaling method and estimates of the approximations Elements of hydrostatics The convection problem. Rayleigh's equations Kinematics: incompressible fields, vector potentials, decompositions of a general field Vorticity conservation in Euler equation. Clebsch potentials and Hamiltonian form of Euler equations. Bidimensional fiuids Empirical algorithms. Analytical theories Incompressible Euler and Navier-Stokes fiuidodynamics. First empirical solutions algorithms. Auxiliary friction and heat equation comparison methods Another class of empirical algorithms. Spectral method. Stokes problem. Gyroscopic analogy Vorticity algorithms for incompressible Euler and Navier-Stokes fiuids. The d = 2 case Vorticity algorithms for incompressible Euler and Navier-Stokes fiuids. The d = 3 case Analytical theories and mathematical aspects Spectral method and local existence, regularity and uniqueness theorems for Euler and Navier-Stokes equations, d Weak global existence theorems for NS. Autoregularization, existence, regularity and uniqueness for d = Regularity: partial results for the NS equation in d = The theory of Leray Fractal dimension of singularities of the Navier-Stokes equation, d = Local homogeneity and regularity. CKN theory Incipient turbulence and chaos Fluids theory in absence of existence and uniqueness theorems for the basic fiuidodynamics equations. Truncated NS equations. The Rayleigh's and Lorenz' models Onset of chaos. Elements of bifurcation theory Chaos scenarios Dynamical tables Ordering chaos Quantitative description of chaotic motions before developed turbulence. Continuous spectrum Timed observations. Random data Dynamical systems types. Statistics on attracting sets Dynamical bases and Lyapunov exponents SRB Statistics. Attractors and attracting sets. Fractal dimension Ordering of Chaos. Entropy and complexity Symbolic dynamics. Lorenz model. Ruelle's principle Developed turbulence Functional integral representation of stationary distributions Phenomenology of developed turbulence and Kolmogorov laws he shell model. Multifractal statistics Statistical properties of turbulence Viscosity, reversibility and irreversible dissipation Reversibility, axiom C, chaotic hypothesis Chaotic hypothesis, fluctuation theorem and Onsager reciprocity. Entropy driven intermittency The structure of the attractor for the Navier-Stokes equations Bibliography Author index Subject index Citations index
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