Diagram equations of the theory of fully developed turbulence

Skeleton diagram equations of turbulence theory — the Dyson equations and the equations for vertices of three types — are obtained nonperturbatively. Their derivation is based on the use of an equation in functional derivatives for the characteristic functional of a hydrodynamic system described by Navier-Stokes equations in the presence of an external random force. The iterative solution of these equations reproduces the perturbation series for second moments that is usually obtained in a more complicated way and also the series for the third moments.Institute of Problems in Mechanics, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 28–37, October, 1994.     

Scale-invariant asymptotic (automodel) behavior in quantum field theory

We summarize the results of our 40-year investigations of scale-invariant (automodel) behavior of form factors in hadron-lepton deep-inelastic scattering processes at high energies and large transferred momenta within the Bogoliubov axiomatics of quantum field theory. This approach was conducive to the emergence of the notion of quark color in quantum chromodynamics.     

The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler _c(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr _c(t) and the pulsation energye(t) it predicts the power lawsr _c(t)t^2/5 andr(t)t ^–6/5.Physics Research Institute of the St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 150–159, July, 1993.     

Renormalizing approach to the theory of developed turbulence: Infrared asymptotic of scaling functions

The problem of infrared discrepancies of the velocity correlation functions in stochastic hydrodynamics is examined within the framework of the quantum field renormalization group method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 18–28, 1988.     

Renormalization group in the theory of fully developed turbulence. Composite operators of canonical dimension 8

Renormalization and critical dimensions of the family of Galilean invariant scalar composite operators of canonical dimension eight are considered within the framework of the renormalization group approach to the stochastic theory of fully developed turbulence.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 92–101, January, 1996.     

Wavelet expansions and asymptotic behavior of distributions

We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line S₀(R)⊂S(R) and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients.     

Renormalization group in the theory of fully developed turbulence. The problem of infrared-essential corrections to the Navier-Stokes equation

Using the RG approach to the theory of fully developed turbulence, we consider the problem of possible IR-essential corrections to the Navier-Stokes equation. We formulate an exact criterion for the actual IR-essentiality of the corrections. In accordance with this criterion. we check whether certain classes of composite operators are IR-essential. All of these operators turn out to be actually IR-inessential for arbitrary values of the RG expansion parameter . This confirms the absence of the crossover and enables the RG results obtained for asymptotically small values of to be extrapolated to the physical range >2.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 47–63, April, 1996.Translated by M. V. Chekhova.     

Renormalization group in the problem of fully developed turbulence of a compressible fluid

A model of fully developed turbulence of a compressible liquid (gas), based on the stochastic Navier-Stokes equation, is considered by means of the renormalization group. It is proved that the model is multiplicatively renormalized in terms of the “velocity-logarithm of density” variables. The scaling dimensions of the fields and parameters are calculated in the one-loop approximation. Dependence of the effective sound velocity and the Mach number on the integral turbulence scale L is studied. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 385–398, March, 1997.     

Numerical analysis of convergent perturbation expansions in quantum theory

We present the results of a numerical analysis of the convergence of the new perturbation expansion recently proposed by Belokurov, Solovyev, and Shavgulidze. Two particular examples are considered: the anharmonic oscillator in quantum mechanics and the renormalization group β-function in field theory. It is shown that in the first case, the series converges to an exact value in a wide range of expansion parameters. This range can be enlarged with the help of the Padé approximation. In field theory, the results have a stronger dependence on the regularization parameter. We discuss an algorithm for choosing this parameter that produces stable results. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 291–297, February, 1997.     

Functional approach to the theory of two-dimensional quantum Fermi systems in an external periodic field

The behavior of an adsorbed layer of Fermi particles in a weak external field with symmetry of the centered quadratic lattice is investigated. Instability of the weakly inhomogeneous state of the system with respect to small fluctuations in the vicinity ofa ^*=2k _F is established; herea ^* is the magnitude of the vectors of the neighbors closest to the point in the inverse lattice; k_F is the Fermi momentum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 29–43, 1988.It remains to thank V. N. Popov for constant interest in the work.     

Ultraviolet problems in field theory and multiscale expansions

An introductory survey is given of ultraviolet problems in Euclidean quantum field theory which are heuristically interpreted either with the aid of the classical renormalization theory or with the aid of Wilson's renormalization group strategy. A unification of each of these approaches with the method of multiscale cluster expansions is necessary for strict proofs.Translated from Itogi Nauki i Tekhniki, Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 24, pp. 111–189, 1986.     

Some results in asymptotic field point theory

In 1944, Levinson ([22]) introduced the concept of dissipativeness for a map T in a finite-dimensional space which leads to the existence of a fixed point of some iterate T ⁿ for n large, rather than a fixed point of T. Browder ([3]) gave an asymptotic field point theorem which proved that T itself had a field point. Although Browder’s result was a big step, it was not suitable for hyperbolic PDEs and neutral functional differential equations because, in those cases, the map T is not compact. For α-contraction maps the result was extended by Nussbaum ([25]) and Hale and Lopes ([13]) using different methods. In this paper, we review these ideas and some more recent applications. Dedicated to Felix Browder on the occasion of his 80th birthday     

An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities

We develop an asymptotic theory of nonlinear operator differential equations of an arbitrary order in Banach spaces. The nonlinear part of the equation is written in a divergent form. It is shown that the main term in an asymptotic representation of solutions at infinity satisfies a finite-dimensional dynamical system perturbed by a small nonlocal operator.     

Analytic theory of finite asymptotic expansions in the real domain. Part I: two-term expansions of differentiable functions

We establish a general analytic theory of asymptotic expansions of type 1 $$f(x) = a_1 \varphi _1 (x) + \cdots + a_n \varphi _n (x) + o(\varphi _n (x)) x \to x_0 ,$$ , where the given ordered n-tuple of real-valued functions (? ₁, ..., ? _n ) forms an asymptotic scale at x ₀ ?? . By analytic theory, as opposed to the set of algebraic rules for manipulating finite asymptotic expansions, we mean sufficient and/or necessary conditions of general practical usefulness in order that (*) hold true. Our theory is concerned with functions which are differentiable (n ? 1) or n times and the presented conditions involve integro-differential operators acting on f, ? ₁, ..., ? _n . We essentially use two approaches; one of them is based on canonical factorizations of nth-order disconjugate differential operators and gives conditions expressed as convergence of certain improper integrals, very useful for applications. The other approach starts from simple geometric considerations and gives conditions expressed as the existence of finite limits, as x ?? x ₀, of certain Wronskian determinants constructed with f, ? ₁, ..., ? _n . There is a link between the two approaches and it turns out that some of the integral conditions found via the factorizational approach have geometric meanings. Our theory extends to more general expansions the theory of real-power asymptotic expansions thoroughly investigated in previous papers. In the first part of our work we study the case of two comparison functions ? ₁, ? ₂ because the pertinent theory requires a very limited theoretical background and completely parallels the theory of polynomial expansions.