Langevin equation of a fluid particle in wall-induced turbulence

We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-induced turbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer). The analysis is based on the asymptotic behavior at a large Reynolds number. We use the Lagrangian Kolmogorov theory, recently derived asymptotic expressions for the spatial distribution of turbulent energy dissipation, and also newly derived reciprocity relations analogous to the Onsager relations supplemented with recent measurement results. The long-time limit of the derived Langevin equation yields the diffusion equation for admixture dispersion in wall-induced turbulence.     

Composite operators,operator expansion,and Galilean invariance in the theory of fully developed turbulence. Infrared corrections to Kolmogorov scaling

The quantum-field renormalization group and operator expansion are used to investigate the infrared asymptotic behavior of the velocity correlation function in the theory of fully developed turbulence. The scaling dimensions of all composite operators constructed from the velocity field and its time derivatives are calculated, and their contributions to the operator expansion are determined. It is shown that the asymptotic behavior of the equal-time correlation function is determined by Galilean-invariant composite operators. The corrections to the Kolmogorov spectrum associated with the operators of canonical dimension 6 are found. The consequences of Galilean invariance for the renormalized composite operators are considered.State University, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 3, pp. 382–401, September, 1994.     

Corrections to fully developed turbulent spectra due to compressibility of the fluid

We construct the regular expansion at small compressibilities for the theory of fully developed turbulence of an isotropic homogeneous compressible fluid with MSR-type action. The parameter of the expansion is the Mach numberMa. For the inertial range of a compressible fluid, we study the infrared singularities determined by the transverse fields, which are used in the theory of incompressible fluids. These singularities are connected with the composite operators of transverse fields that are investigated by the quantum field renormalization group method. As a result, it is shown that the transverse fields induce scaling behavior with theMa scaling dimension equal to 1/3 (i.e.,Ma k^–1/3 is the dimensionless scaling parameter of the correlation functions in the inertial range).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 375–389, March, 1996.Translated by L. O. Chekhov.     

Renormalization group in the theory of turbulence: Three-loop approximation as <Emphasis Type="Italic">d</Emphasis> → ∞

We use the renormalization group method to study the stochastic Navier-Stokes equation with a random force correlator of the form k ^4−d−2ɛ in a d-dimensional space in connection with the problem of constructing a 1/d-expansion and going beyond the framework of the standard ɛ-expansion in the theory of fully developed hydrodynamic turbulence. We find a sharp decrease in the number of diagrams of the perturbation theory for the Green’s function in the large-d limit and develop a technique for calculating the diagrams analytically. We calculate the basic ingredients of the renormalization group approach (renormalization constant, β-function, fixed-point coordinates, and ultraviolet correction index ω) up to the order ɛ ³ (three-loop approximation). We use the obtained results to propose hypothetical exact expressions (i.e., not in the form of ɛ-expansions) for the fixed-point coordinate and the index ω. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 460–477, March, 2009.     

A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications

We obtain a differential analog of the main lemma of the theory of Markov branching processes μ(t), t ≥ 0, with continuous time. We show that the results obtained can be used in the proof of limit theorems of the theory of branching processes by the known Stein-Tikhomirov method. Moreover, in contrast to the classical condition of nondegeneracy of the branching process {μ(t) > 0}, we consider the condition of its nondegeneracy in the distant future {μ(∞) > 0} and justify it in terms of generating functions. Under this condition, we study the asymptotic behavior of the trajectory of the process considered.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 258–264, February, 2005.     

Two Dimensional Incompressible Ideal Flow Around a Small Obstacle

Abstract

In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on γ, the circulation around the obstacle. For smooth flow around a single obstacle, γ is a conserved quantity which is determined by the initial data. We will show that if γ = 0, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if γ≠ 0, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

Does there exist a lebesgue measure in the infinite-dimensional space?

We consider sigma-finite measures in the space of vector-valued distributions on a manifold X with the characteristic functional Ψ(f) = exp{−θ ∫_X ln‖f(x)‖dx}, θ > 0. The collection of such measures constitutes a one-parameter semigroup relative to θ. In the case of scalar distributions and θ = 1, this measure may be called the infinite-dimensional Lebesgue measure. We prove that the weak limit of the Haar measures on the Cartan subgroups of the groups SL(n, ℝ), when n tends to infinity, is that infinite-dimensional Lebesgue measure. This measure is invariant under the linear action of some infinite-dimensional abelian group that can be viewed as an analog of an infinite-dimensional Cartan subgroup; this fact can serve as a justification of the name Lebesgue as a valid name for the measure in question. Application to the representation theory of current groups was one of the reasons to define this measure. The measure is also closely related to the Poisson-Dirichlet measures well known in combinatorics and probability theory. The only known example of analogous asymptotic behavior of the uniform measure on the homogeneous manifold is the classical Maxwell-Poincaré lemma, which states that the weak limit of uniform measures on the Euclidean spheres of appropriate radius, as dimension tends to infinity, is the standard infinite-dimensional Gaussian measure. Our situation is similar, but all the measures are no more finite but sigma-finite. The result raises an important question about the existence of other types of interesting asymptotic behavior of invariant measures on the homogeneous spaces of Lie groups.     

Asymptotic Analysis of a Nonlinear Parabolic Problem Modelling Miscible Compressible Displacement in Porous Media

We study the asymptotic behavior, with respect to high Peclet numbers, of a model describing a compressible and miscible displacement in a porous medium. The transport of mass is then described by a nonlinear, fully coupled and degenerate parabolic system. Using non-classical estimates and renormalization tools, we prove existence of relevant weak solutions for the limit problem.      

Asymptotic Limit of the Bayes Actions Set Derived from a Class of Loss Functions

Similarly to the determination of a prior in Bayesian Decision theory, an arbitrarily precise determination of the loss function is unrealistic. Thus, analogously to global robustness with respect to the prior, one can consider a set of loss functions to describe the imprecise preferences of the decision maker. In this paper, we investigate the asymptotic behavior of the Bayes actions set derived from a class of loss functions. When the collection of additional observations induces a decrease in the range of the Bayes actions, robustness is improved. We give sufficient conditions for the convergence of the Bayes actions set with respect to the Hausdorff metric and we also give the limit set. Finally, we show that these conditions are satisfied when the set of decisions and the set of states of nature are subsets of ^p.     

Does there exist a lebesgue measure in the infinite-dimensional space?

We consider sigma-finite measures in the space of vector-valued distributions on a manifold X with the characteristic functional Ψ(f) = exp{−θ ∫_X ln‖f(x)‖dx}, θ > 0. The collection of such measures constitutes a one-parameter semigroup relative to θ. In the case of scalar distributions and θ = 1, this measure may be called the infinite-dimensional Lebesgue measure. We prove that the weak limit of the Haar measures on the Cartan subgroups of the groups SL(n, ℝ), when n tends to infinity, is that infinite-dimensional Lebesgue measure. This measure is invariant under the linear action of some infinite-dimensional abelian group that can be viewed as an analog of an infinite-dimensional Cartan subgroup; this fact can serve as a justification of the name Lebesgue as a valid name for the measure in question. Application to the representation theory of current groups was one of the reasons to define this measure. The measure is also closely related to the Poisson-Dirichlet measures well known in combinatorics and probability theory. The only known example of analogous asymptotic behavior of the uniform measure on the homogeneous manifold is the classical Maxwell-Poincaré lemma, which states that the weak limit of uniform measures on the Euclidean spheres of appropriate radius, as dimension tends to infinity, is the standard infinite-dimensional Gaussian measure. Our situation is similar, but all the measures are no more finite but sigma-finite. The result raises an important question about the existence of other types of interesting asymptotic behavior of invariant measures on the homogeneous spaces of Lie groups. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 256–281.     

A central limit theorem for extremal characters of the infinite symmetric group

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.     

Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Time asymptotics for solutions of the Burgers equation with a periodic force

We consider the Burgers equation with a periodic force which presents a simplified model for turbulence. We are interested in the asymptotic behaviour of solutions for . This problem has been studied by Sinai who uses a probabilistic and very technical approach. Using methods from spectral theory we get similar results. This functional analytic approach gives an easier proof. For certain initial data (periodic or some random perturbations of those) we show time-convergence towards a deterministic periodic limit solution related to the ground state of a certain Schr?dinger operator. Received June 10, 1998     

A central limit theorem and higher order results for the angular bispectrum

The angular bispectrum of spherical random fields has recently gained an enormous importance, especially in connection with statistical inference on cosmological data. In this paper, we analyze its moments and cumulants of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations. The results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behavior of coefficients arising in matrix representation theory for the group of rotations SO(3). I am very grateful to an associate editor and two referees for many useful comments, and to M. W. Baldoni and P. Baldi for discussions on an earlier version.     

Almost sure local limit theorem for the Dickman distribution

We study the asymptotic behavior, and more precisely the second order properties, of the probabilistic model introduced in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002) for describing the Dickman distribution. This model appears as an extremal example in the theory of the local and almost sure local limit theorem. We establish a delicate correlation inequality for this system. We apply it to obtain a fine almost sure local limit theorem. In doing so, we also give a corrected proof of the corresponding local limit theorem stated in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002).     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

Periodic Operators in High-Contrast Media and the Integrated Density of States Function

ABSTRACT

This paper studies the asymptotic behavior for the integrated density of states function for operators associated with the propagation of classical waves in a high-contrast, periodic, two-component medium. Consider a domain Ω₊ contained in the hypercube [0, 2π) ⁿ . We define a function χ_τ which takes the value 1 in Ω₊ and the value τ in [0, 2π) ⁿ \ Ω₊. We extend this setup periodically to ? ⁿ and define the operator L _τ = ??χ_τ ?. As τ goes to infinity, it is known that the spectrum of L _τ exhibits a band-gap structure and that the spectral density accumulates at the upper endpoints of the bands. We establish the existence and some important properties of a rescaled integrated density of states function in the large coupling limit which describes the non-trivial asymptotic behavior of this spectral accumulation.