[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

We consider the vanishing viscosity limit of the Navier--Stokes equations in a half-plane, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier--Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier--Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.

Remarks on the Inviscid Limit for the Navier--Stokes Equations for Uniformly Bounded Velocity Fields

We present a simple, short and elementary proof that if v is a Beltrami flow with a finite energy in R3 then v = 0. In the case of the Beltrami flows satisfying v ∈ Lloc(R) ∩ Lq(R3) with q ∈ [2, 3), or |v(x)| = O(1/|x|1+e) for some e > 0, we provide a different, simple proof that v = 0. AMS Subject Classification Number: 35Q31, 76B03, 76W05 keywords: Euler equations, Beltrami flows, Liouville type theorem

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Remarks on a Liouville-Type Theorem for Beltrami Flows

We prove global existence and uniqueness of solutions of Oldroyd-B systems with relatively small data in $\Rr^d$, in a large functional setting ($C^{\alpha}\cap L^1$). This is a stability result, solutions select an equilibrium and converge exponentially to it. Large spatial derivatives of the initial density and stress are allowed, provided the $L^{\infty}$ norm of the density and stress are small enough. We prove global regularity for large data for a model in which the potential responds to high rates of strain in the fluid. We also prove global existence for a class of large data for a didactic scalar model which attempts to capture in the simplest way the essence of the dissipative nature of the coupling to fluid. This latter model has an unexpected cone invariance in function space that is crucial for the global existence.

Remarks on Oldroyd-B and Related Complex Fluids Models

Inelastic collisions of three particles on a line as a two-dimensional billiard

We prove global existence and uniqueness of solutions of Oldroyd-B systems with relatively small data in R d , in a large functional setting (C � \ L 1 ). This is a stability result, solutions select an equilibrium and converge exponentially to it. Large spatial derivatives of the initial density and stress are allowed, provided the L 1 norm of the density and stress are small enough. We prove global regularity for large data for a model in which the potential responds to high rates of strain in the fluid. We also prove global existence for a class of large data for a didactic scalar model which attempts to capture in the simplest way the essence of the dissipative nature of the coupling to fluid. This latter model has an unexpected cone invariance in function space that is crucial for the global existence. 1. Introduction The complex fluids models we consider treat the interaction of a normal fluid (incompressible in our case) with insoluble matter suspended in it. Models have been devised to deal with microscopic elastic thread-like objects such as polymers ((12), (26)). The complicated hydrodynamic interactions are simplified using the separation of scales, replacing the many degrees of freedom due to them by few representative ones. In the models we consider here, an end-to-end vector in R 3 (m in this paper) represents the orientation of the polymer and it is the sole variable retained to describe the microscopic object. The models that are most studied with this degree of simplification include the kinetic description of the Oldroyd B model, the FENE model and variants. The probability distribution of the vectors m depends on time and physical space, and it is assumed to be absolutely continuous with respect to the usual Lebesgue measure dm, so the chance of finding m at time t and location x 2 R 3 in the volume element dm is f(x;m;t)dm. The aim is to describe the evolution of f in space and time. When the polymer concentration is so dilute that the polymers do not interact, and when the fluid is smooth, then the equation of f is a linear Fokker-Planck equation. The effect of the fluid can be understood perturbatively: because the objects suspended are assumed to have faster time scales and shorter length scales than the scales on which solvent varies, it is then justified to treat the fluid as varying little as far as the microscopic suspended objects are concerned. This macro-micro interaction can be rationally discussed, and leads to a kinetic description of the particles, given a fluid flow. On the other hand, the collective effect that the particles have on the flow itself is a macroscopic effect of microscopic insertions: a micro- macro interaction. These interactions are much more mysterious. There have been attempts to produce systematic upscalings based on non-equilibrium thermodynamics formalisms ((2), (13)). When the microscopic insertions have a larger number of degrees of freedom, m 2 M, where M is a Riemannian manifold representing a finite number of degrees of freedom with constrains, then the kinetic equation is a Smoluchowski equation on the manifold. The upscaling principle advocated by one of us ((5), (6)) is easiest formulated as the requirement that the sum of the energy of the fluid

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Peter Constantin

Papers

Remarks on the Inviscid Limit for the Navier--Stokes Equations for Uniformly Bounded Velocity Fields

Remarks on a Liouville-Type Theorem for Beltrami Flows

Remarks on Oldroyd-B and Related Complex Fluids Models

Inelastic collisions of three particles on a line as a two-dimensional billiard

Remarks on oldroyd-b and related complex fluid models