[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

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Non-planar fronts in Boussinesq reactive flows

Necessary and sufficient conditions for the absence of singularities in solutions of the three dimensional Navier-Stokes equations are recalled. New global weak solutions are constructed. They enjoy the properties that the spatial integral of the vorticity magnitude is a priori bounded in time and that the space and time integral of the $\frac{4}{3+\varepsilon}$ power of the magnitude of the gradient of the vorticity is a priori bounded. Vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. The two dimensional Hausdorff measure of level sets of the vorticity magnitude is studied and a priori bounds on an average such measure, are obtained. When expressed in terms of the Reynolds number and the Kolmogorov dissipation length η, these bounds are 

$$ \leq \frac{L^3}{\eta}(1+ Re^{-\frac{1}{2}})^{\frac{1}{2}}$$

The area of level sets of scalara and in particular isotherms in Rayleigh-Benard convection is studied. A quantity, r,t (x 0), describing an average value of the area of a portion of a level set contained in a small ball of radius r about the point x 0 is bounded from above by 

$$ _{r,t}(x_0)\leq C \kappa^{-\frac{1}{2}}r^{\frac{5}{2}} ^{\frac{1}{2}}$$

where κ is the diffusivity constant and is the average maximal velocity. The inequality is valid for r larger than the local length scale $\lambda = \kappa ^{-1}$. It is found that this local scale is a microscale, if the statistics of the velocity field are homogeneous. This suggests that 2.5 is a lower bound for the fractal dimension of an (ensemble) average interface in homogeneous turbulent flow, a fact which agrees with the experimental lower bound of 2.35.

Remarks on the Navier-Stokes Equations

We consider systems of reactive Boussinesq equations in two-dimensional strips that are not aligned with gravity's direction. We prove that for any width of such strips and for arbitrary Rayleigh and Prandtl numbers, the systems admit smooth, non-planar travelling wave solutions with the fluid's velocity satisfying no-slip boundary conditions.

Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions

Using the methods of Foias [Sem. Math. Univ. Padova 48, 219–343 (1972); 49, 9–123 (1973)] and Vishik–Fursikov [Mathematical Problems of Statistical Hydromechanics (Kluwer, Dordrecht, 1988)], we prove the existence and uniqueness of both spatial and space–time statistical solutions of the Navier–Stokes equations on the phase space of vorticity. Here the initial vorticity is in Yudovich space and the initial measure has finite mean enstrophy. We show under further assumptions on the initial vorticity that the statistical solutions of the Navier–Stokes equations converge weakly and the inviscid limits are the corresponding statistical solutions of the Euler equations.

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Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits

We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.

Peter Constantin

Papers

Non-planar fronts in Boussinesq reactive flows

Remarks on the Navier-Stokes Equations

Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions

Statistical solutions of the Navier-Stokes equations on the phase space of vorticity and the inviscid limits

Remarks on high Reynolds numbers hydrodynamics and the inviscid limit