Foreword
Preface
1 Mathematical Background
1.1 Dynamical systems
1.1.1 Vector felds and dynamical systems
1.1.2 Critical points in phase space
1.1.3 Higher-order autonomous systems
1.1.4 Dirac delta function
1.1.5 Special functions
1.1.6 Green's function
1.1.7 Boundary and initial value problems
1.2 Asymptotic behavior and stability
1.2.1 Asymptotic expansions
1.2.2 Asymptotic behavior of autonomous systems
1.2.3 Stability of autonomous systems
1.2.4 More on stability
1.3 Bifurcations
1.3.1 Instability and bifurcations
1.3.2 Saddle-node bifurcation
1.3.3 Transcritical and pitchfork bifurcations
1.3.4 Hopf bifurcation
1.3.5 Saddle-node bifurcation of a periodic orbit
1.3.6 Global bifurcation
1.4 Attractors
1.4.1 Chaotic motion and symbolic dynamics
1.4.2 Homoclinic tangles and Smale's horseshoe map
1.4.3 Poincaré return map
1.4.4 Lyapunov's exponents and entropy
1.4.5 Attracting sets and attractors
1.5 Fractals
1.5.1 Local structure of fractals
1.5.2 Operations with fractals
1.5.3 Fractal attractors in dynamical systems
1.6 Perturbations
1.6.1 Regular perturbation theory
1.6.2 Singular perturbation theory
1.7 Elements of tensor analysis
1.7.1 Transformations of coordinate systems
1.7.2 Covariant and contravariant derivatives
1.7.3 Christoffel symbols and curvature tensor
1.7.4 Integral formulas
1.8 Navier-Stokes equations for nonequilibrium gas mixture
1.8.1 Continuity,momentum and energy equations
1.8.2 Closing relations and transport coefficients
1.8.3 Boundary conditions
1.8.4 Deducing Navier-Stokes equation
1.8.5 Existence and uniqueness of solutions of the Navier—Stokes equation
1.8.6 Relativistic Navier—Stokes equation
1.9 Exercises
bliography
2 Models for Hydrodynamic Instabilities
2.1 Stability concepts
2.1.1 Boundary conditions
2.1.2 Inviscid and high-Reynolds—number flow
2.1.3 Basic definitions
2.2 Rayleigh—Taylor instability
2.2.1 Potential flow
2.2.2 Plane boundaries
2.2.3 Spherical boundaries
2.2.4 Nonlinear perturbation theory
2.2.5 Inhomogeneous fluids
2.2.6 Ⅵscous fluids
2.3 Kelvin-Helmholtz instability
2.3.1 Instability of annular incompressible jet
2.3.2 Rotating jets
2.3.3 Supersonic viscous jet
2.3.4 Supersonic viscous jet with Gaussian sound velocity distribution
2.3.5 Relativistic jet
2.4 Exercises
bliography
3 Models for Turbulence
3.1 Symmetries and conservation laws
3.1.1 Euler and Navier—Stokes equations
3.1.2 Symmetries
3.1.3 Conservation laws
3.2 Anomalous scaling exponents
3.2.1 Multifractal models
3.2.2 Random variables and correlation functions
3.2.3 Richardson-Kolmogorov concept of turbulence
3.2.4 Scaling of the structure hmctions
3.2.5 Dissipative and dynamical scaling
3.2.6 Fusion rules in turbulence systems
3.3 Calculation of scaling exponents
3.3.1 Basic formulas
3.4 Bifurcations for the Kuramoto-Sivashinsky equation
3.4.1 Symmetry：translations, reflections, and O(2)-equivariance
3.4.2 Kuramoto-Sivashinsky equation
3.5 Strange attractors and turbulence
3.5.1 The Taylor—Couette experiment
3.5.2 Dynamical systems with one observable
3.5.3 Limit capacity and dimension
3.5.4 Dimension and entropy
3.6 Global attractor for Navier-Stokes equation
3.6.1 The ladder inequality
3.6.2 Estimates
3.6.3 Length scales in the two-dimensional case
3.6.4 Three-dimensional regularity
3.6.5 The attractor dimension
3.7 Hierarchical sheU models
3.7.1 Gledzer-Ohkitani-Yamada shell model
3.7.2 (N，￡)-sabra shell models
3.7.3 Navier-Stokes equations in the common wavelets repre
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