ForewordPreface1 Mathematical Background1.1 Dynamical systems1.1.1 Vector felds and dynamical systems1.1.2 Critical points in phase space1.1.3 Higher-order autonomous systems1.1.4 Dirac delta function1.1.5 Special functions1.1.6 Green‘s function1.1.7 Boundary and initial value problems1.2 Asymptotic behavior and stability1.2.1 Asymptotic expansions1.2.2 Asymptotic behavior of autonomous systems1.2.3 Stability of autonomous systems1.2.4 More on stability1.3 Bifurcations1.3.1 Instability and bifurcations1.3.2 Saddle-node bifurcation1.3.3 Transcritical and pitchfork bifurcations1.3.4 Hopf bifurcation1.3.5 Saddle-node bifurcation of a periodic orbit1.3.6 Global bifurcation1.4 Attractors1.4.1 Chaotic motion and symbolic dynamics1.4.2 Homoclinic tangles and Smale’s horseshoe map1.4.3 Poincaré return map1.4.4 Lyapunovs exponents and entropy1.4.5 Attracting sets and attractors1.5 Fractals1.5.1 Local structure of fractals1.5.2 Operations with fractals1.5.3 Fractal attractors in dynamical systems1.6 Perturbations1.6.1 Regular perturbation theory1.6.2 Singular perturbation theory1.7 Elements of tensor analysis1.7.1 Transformations of coordinate systems1.7.2 Covariant and contravariant derivatives1.7.3 Christoffel symbols and curvature tensor1.7.4 Integral formulas1.8 Navier-Stokes equations for nonequilibrium gas mixture1.8.1 Continuity，momentum and energy equations1.8.2 Closing relations and transport coefficients1.8.3 Boundary conditions1.8.4 Deducing Navier-Stokes equation1.8.5 Existence and uniqueness of solutions of the Navier—Stokes equation1.8.6 Relativistic Navier—Stokes equation1.9 Exercisesbliography2 Models for Hydrodynamic Instabilities2.1 Stability concepts2.1.1 Boundary conditions2.1.2 Inviscid and high-Reynolds—number flow2.1.3 Basic definitions2.2 Rayleigh—Taylor instability2.2.1 Potential flow2.2.2 Plane boundaries2.2.3 Spherical boundaries2.2.4 Nonlinear perturbation theory2.2.5 Inhomogeneous fluids2.2.6 Ⅵscous fluids2.3 Kelvin-Helmholtz instability2.3.1 Instability of annular incompressible jet2.3.2 Rotating jets2.3.3 Supersonic viscous jet2.3.4 Supersonic viscous jet with Gaussian sound velocity distribution2.3.5 Relativistic jet2.4 Exercisesbliography3 Models for Turbulence3.1 Symmetries and conservation laws3.1.1 Euler and Navier—Stokes equations3.1.2 Symmetries3.1.3 Conservation laws3.2 Anomalous scaling exponents3.2.1 Multifractal models3.2.2 Random variables and correlation functions3.2.3 Richardson-Kolmogorov concept of turbulence3.2.4 Scaling of the structure hmctions3.2.5 Dissipative and dynamical scaling3.2.6 Fusion rules in turbulence systems3.3 Calculation of scaling exponents3.3.1 Basic formulas3.4 Bifurcations for the Kuramoto-Sivashinsky equation3.4.1 Symmetry：translations， reflections， and O(2)-equivariance3.4.2 Kuramoto-Sivashinsky equation3.5 Strange attractors and turbulence3.5.1 The Taylor—Couette experiment3.5.2 Dynamical systems with one observable3.5.3 Limit capacity and dimension3.5.4 Dimension and entropy3.6 Global attractor for Navier-Stokes equation3.6.1 The ladder inequality3.6.2 Estimates3.6.3 Length scales in the two-dimensional case3.6.4 Three-dimensional regularity3.6.5 The attractor dimension3.7 Hierarchical sheU models3.7.1 Gledzer-Ohkitani-Yamada shell model3.7.2 (N，￡)-sabra shell models3.7.3 Navier-Stokes equations in the common wavelets repre
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