[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

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

It has often been remarked that turbulence is a subject of great scientific and technological importance, and yet one of the least understood (e.g. McComb 1990). To an outsider this may seem strange, since the basic physical laws of fluid mechanics are well established, an excellent mathematical model is available in the Navier-Stokes equations, and the results of well over a century of increasingly sophisticated experiments are at our disposal. One major difficulty, of course, is that the governing equations are nonlinear and little is known about their solutions at high Reynolds number, even in simple geometries. Even mathematical questions as basic as existence and uniqueness are unsettled in three spatial dimensions (cf Temam 1988). A second problem, more important from the physical viewpoint, is that experiments and the available mathematical evidence all indicate that turbulence involves the interaction of many degrees of freedom over broad ranges of spatial and temporal scales. One of the problems of turbulence is to derive this complex picture from the simple laws of mass and momentum balance enshrined in the NavierStokes equations. It was to this that Ruelle & Takens (1971) contributed with their suggestion that turbulence might be a manifestation in physical

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The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

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

Both an original contribution and a lucid introduction to mathematical aspects of fluid mechanics, Navier-Stokes Equations provides a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.

Navier-Stokes equations

The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments. This active scalar represents the temperature evolving on the two dimensional boundary of a rapidly rotating half space with small Rossby and Ekman numbers and constant potential vorticity. The possibility of frontogenesis within this approximation is an important issue in the context of geophysical flows. A striking mathematical and physical analogy is developed between the structure and formation of singular solutions of this quasi-geostrophic active scalar in two dimensions and the potential formation of finite time singular solutions for the 3-D Euler equations. Detailed mathematical criteria are developed as diagnostics for self-consistent numerical calculations indicating strong front formation. These self-consistent numerical calculations demonstrate the necessity of nontrivial topology involving hyperbolic saddle points in the level sets of the active scalar in order to have singular behaviour; this numerical evidence is strongly supported by mathematical theorems which utilize the nonlinear structure of specific singular integrals in special geometric configurations to demonstrate the important role of nontrivial topology in the formation of singular solutions.

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Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.

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Onsager's conjecture on the energy conservation for solutions of Euler's equation

Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone Invariance Properties.- Consequences Regarding the Global Attractor.- Local Exponential Decay Toward Blocked Integral Surfaces.- Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- Choice of the Initial Manifold.- Construction of the Inertial Mainfold.- Lower Bound for the Exponential Rate of Convergence to the Attractor.- Asymptotic Completeness: Preparation.- Asymptotic Completeness: Proof of Theorem 12.1.- Stability with Respect to Perturbations.- Application: The Kuramoto-Sivashinsky Equation.- Application: A Nonlocal Burgers Equation.- Application: The Cahn-Hilliard Equation.- Application: A parabolic Equation in Two Space Variables.- Application: The Chaffee-Infante Reaction Diffusion Equation.- References.- Index.

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data? For the linear wave equation, for instance, the answer is no. However, in [10] T. Kato found a local smoothing property of the Korteweg-de Vries equation: the solution of the initial value problem is, locally, one derivative smoother than the initial datum. Kato's proof uses, in a curcial way, the algebraic properties of the symbol for the Korteweg-de Vries equation and the fact that the underlying spatial dimension is one. Actually, judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect, one would be inclined to believe this was a special property of the K-dV equation. This is not, however, the case. In this paper, we attempt to describe a general local smoothing effect for dispersive equations and systems. The smoothing effect is due to the dispersive nature of the linear part of the equation. All the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property. To mention only a few, the K-dV, Benjamin-Ono, intermediate long wave, various Boussinesq, and Schrodinger equations are included. We study, thus, equations and systems of the form

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Peter Constantin

Papers

Navier-Stokes equations

Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

Onsager's conjecture on the energy conservation for solutions of Euler's equation

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Local smoothing properties of dispersive equations