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Turbulence and the dynamics of coherent structures. I. Coherent structures

Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

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Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Monotone dynamical systems Stability and convergence Competitive and cooperative differential equations Irreducible cooperative systems Cooperative systems of delay differential equations Nonquasimonotone delay differential equations Quasimonotone systems of parabolic equations A competition model Appendix Bibliography.

Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems

Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of the singular point where a drop separates. The increased attention is due to a number of recent and increasingly refined experiments, as well as to a host of technological applications, ranging from printing to mixing and fiber spinning. The description of drop separation becomes possible because jet motion turns out to be effectively governed by one-dimensional equations, which still contain most of the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been observed, which are a nested superposition of singularities of the same universal form. At low viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven by capillary waves. The author reviews the theoretical development of this field alongside recent experimental work, and outlines unsolved problems.

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Nonlinear dynamics and breakup of free-surface flows

Jets, ie collimated streams of matter, occur from the microscale up to the large-scale structure of the universe Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology They also arise from the last but one topology change of liquid masses bursting into sprays Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects

Physics of liquid jets

The Mathematical Theory of Viscous Incompressible Flow

I Preliminary Considerations.- II Equations of Elliptic Type.- III Equations of Parabolic Type.- IV Equations of Hyperbolic Type.- V Some Generalizations.- VI The Method of Finite Differences.

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The boundary value problems of mathematical physics

Preface Part I: Attractors for the semigroups of operators 1. Basic notions 2. Semigroups of class K 3. Semigroups of class AK 4. On dimensions of compact invariant sets Part II: Semigroups generated by evolution equations 5. Introduction to Part II 6. Estimates for the number of determining modes and the fractal dimension of bounded invariant sets for the Navier-Stokes equations 7. Evolution equations of hyperbolic type References Index.

Attractors for Semi-groups and Evolution Equations

In this chapter we shall consider second-order parabolic equations and prove the unique solvability of the initial-boundary value problem in the domains Q T =, (x, t): x ∈ Ω,t ∈ (0, T) for the first, second, and third boundary conditions. We shall assume that the domain Ω is bounded, although all the results, except for the representation of solutions by Fourier series, will be valid for an arbitrary unbounded domain Ω. Moreover, the methods of solution given for bounded Ω are applicable to unbounded Ω (in particular, for Ω = R n ),but need minor modification which we shall point out.

Equations of Parabolic Type

In this section we shall consider linear second-order equations 

$$\begin{array}{*{20}{c}} {Lu = \sum\limits_{{i,j = 1}}^{n} {\frac{\partial }{{\partial {{x}_{i}}}}} ({{a}_{{ij}}}(x){{u}_{{{{x}_{j}}}}} + {{a}_{i}}(x)u(x))} \\ {\quad {\mkern 1mu} + \sum\limits_{{i = 1}}^{n} {{{b}_{i}}} (x){{u}_{{{{x}_{i}}}}} + a(x)u = f(x) + \sum\limits_{{i = 1}}^{n} {\frac{{\partial {{f}_{i}}(x)}}{{\partial {{x}_{i}}}}} ,\quad {{a}_{{ij}}}\left( x \right) = {{a}_{{ji}}}\left( x \right),} \\ \end{array}$$

(1.1)

with real coefficients which satisfy the condition of uniform ellipticity in a bounded domain Ω of Euclidean space R n .

O. A. Ladyzhenskaya

Papers

The Mathematical Theory of Viscous Incompressible Flow

The boundary value problems of mathematical physics

Attractors for Semi-groups and Evolution Equations

Equations of Parabolic Type

Equations of Elliptic Type