Kolmogorov Problem Research Articles

Reduction in Degrees of Freedom for Large-Scale Nonlinear Systems in Computer-Assisted Proofs

In many physical systems, it is important to know the exact trajectory of a solution. Relevant applications include celestial mechanics, fluid mechanics, robotics, etc. For cases where analytical methods cannot be applied, one can use computer-assisted proofs or rigorous computations. One can obtain a guaranteed bound for the solution trajectory in the phase space. The application of rigorous computations poses few problems for low-dimensional systems of ordinary differential equations (ODEs) but is a challenging problem for large-scale systems, for example, systems of ODEs obtained from the discretization of the PDEs. A large-scale system size for rigorous computations can be as small as about a hundred ODE equations because computational complexity for rigorous algorithms is much larger than that for simple computations. We are interested in the application of rigorous computations to the problem of proving the existence of a periodic orbit in the Kolmogorov problem for the Navier–Stokes equations. One of the key issues, among others, is the computation complexity, which increases rapidly with the growth of the problem dimension. In previous papers, we showed that 79 degrees of freedom are needed in order to achieve convergence of the rigorous algorithm only for the system of ordinary differential equations. Here, we wish to demonstrate the application of the proper orthogonal decomposition (POD) in order to approximate the attracting set of the system and reduce the dimension of the active degrees of freedom.

Stochastic estimation of Green’s functions with application to diffusion and advection-diffusion-reaction problems

A stochastic method is described for estimating Green’s functions (GF’s), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF’s, within any geometry, the technique solves the central challenge in obtaining Green’s function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF’s that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green’s function, G(x,t|x′,t′), and the backward Kolmogorov problem governing the transition density, p(x,t|x′,t′), of the stochastic process used in Green’s function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green’s function estimation can be used, and for subsequently estimating the transition density and associated Green’s function. Second, a naive estimator for the transition density is proposed and tested. Third, Green’s function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF’s, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF’s.

The Landau–Kolmogorov problem on a finite interval in the Taikov case

We solve the pointwise Landau–Kolmogorov problem on the interval I=[−1,1] on finding |f(k)(t)|→sup under constraints ‖f‖2⩽1 and ‖f(n)‖2⩽σ, where t∈I and σ>0 are fixed. For n=1 and n=2, we solve the uniform version of the Landau–Kolmogorov problem on the interval I in the Taikov case by proving the Karlin-type conjecture supt∈I|f(k)(t)|=|f(k)(1)| under above constraints. The proof relies on the analysis of the dependence of the norm of the solution to higher-order Sturm–Liouville equation (−1)nu(2n)+λu=−λf with boundary conditions u(s)(−1)=u(s)(1)=0, s=0,1,…,n−1, on non-negative parameter λ, where f is some piece-wise polynomial function. Furthermore, we find sharp inequality ‖f(k)‖∞⩽A‖f‖2+B‖f(n)‖2 with the smallest possible constant A>0 and the smallest possible constant B=B(A) for k∈{n−2,n−1}.

Computer-assisted proofs of the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem

We propose a computer-assisted method to prove the existence of a symmetry-breaking bifurcation point for the Kolmogorov problem. First, we numerically show that a symmetry-breaking bifurcation point exists. Then, according to the symmetric property, we define a symmetric operator. Using this operator, we divide the space into a symmetric space and an antisymmetric space. Then, considering the Reynolds number as a variable, we construct an extended system. We confirm the existence of the symmetry-breaking bifurcation point by computer-assisted proofs of the extended system that satisfies both conditions of a bifurcation theorem. The first condition is that the system has an isolated solution and the second is that a linearized operator is bijective. We numerically construct a set containing solutions that satisfy the hypothesis of Banach’s fixed-point theorem in a certain Sobolev space and thus the first condition is satisfied. The second condition is equivalent to an equation having the unique trivial solution zero. We prove that this condition is equivalent to an inequality.

On Landau-Kolmogorov problem on the interval for absolutely monotonic functions

We solve the Landau-Kolmogorov problem and the Kolmogorov problem for three positive numbers on the class of functions which are absolutely monotonic on a finite interval.

The Kolmogorov Problem on Uniqueness of Probability Solutions of a Parabolic Equation

Abstract—We give a solution to the Kolmogorov problem on uniqueness of probability solutions to a parabolic Fokker–Planck–Kolmogorov equation.

Bimodal vortex solutions on a sphere

Steady and time-periodic solutions of the two-dimensional Navier–Stokes equations are studied on a sphere under steady forcings represented by a single spherical harmonic function. This setting is similar to the Kolmogorov problem of the two-dimensional flows under a sinusoidal forcing on a doubly periodic domain (a flat torus), where either simple steady or time-periodic unimodal solutions have been found even at high Reynolds numbers depending on the forcing profiles. In our spherical case, we investigate the bifurcation structure and find either simple steady or time-periodic bimodal solutions, consisting of only two pairs of positive and negative vortices, depending on the forcing profiles at very high Reynolds numbers.

Numerical Bifurcation Analysis in 3D Kolmogorov Flow Problem

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered in the extended periodic domain. The Galerkin{Fourier method is applied and two problem congurations are considered: the symmetry preserving subset of solutions and the full set of solutions. The bifurcation patterns are revealed through the numerical analysis: the eigenvalues of the linearised perturbed system are analyzed (for stationary and periodic solutions) as well as the phase space trajectories that the system generates. Linear stability of the main solution is analyzed. For cubic domain we detected a fast transition to chaos through double Hopf bifurcation. Then the full bifurcation scenario is analyzed for rectangular parallelepiped with double side stretching. The initial stage of laminar-turbulent transition undergoes supercritical pitchfork bifurcation followed by subcritical Hopf bifurction. The system can either go through the series of cycles in Feigenbaum and Sharkovsky ordering or through the bifurcations on invariant tori up to the three and four dimensional tori. The transition to chaos goes either through the emergence of singular attractors of dimension greater than three or through the resonance of tori.

Nonlinear Dynamics of Laminar-Turbulent Transition in Generalized 3D Kolmogorov Problem for Incompressible Viscous Fluid at Symmetric Solution Subset

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered. The Galerkin-Fourier method is applied and the symmetry preserving subset of solutions is considered. The bifurcation patterns are revealed through the numerical analysis of eigenvalues of the linearized perturbed system from the analytical main stationary solution and through the analysis of phase space trajectories that the system generates. It was found that the initial stage of laminar-turbulent transition undergoes pitchfork bifurcation, through which the system can either go through the series of cycle cascades or through continuous tori bifurcations in accordance with the FShM scenario.

The Landau–Kolmogorov problem for the Laplace operator on a ball

In this paper we solve the problem of maximizing the value of the Laplace operator at the origin for functions such that the second degree of the Laplace operator belongs to the space L∞ on the unit ball of the Euclidean space. The problem is solved under restrictions on the uniform norm of a function and the L∞-norm of the second degree of the Laplace operator of this function.

Задача Колмогорова про існування абсолютно монотонної і кратно монотонної функції з заданими нормами похідних

Задача Колмогорова про існування абсолютно монотонної і кратно монотонної функції з заданими нормами похідних

Qualitative analysis of dynamics in Kolmogorov’s problem on a flow of a viscous incompressible fluid

We numerically analyze the qualitative behavior of solutions in the two-dimensional problem on a periodic flow of a viscous incompressible fluid with an exciting force (the Kolmogorov problem). To construct an approximate solution, we use the Bubnov–Galerkin method based on spline approximations. We prove the convergence of the method to the solution of the problem in the weak sense. We compare the obtained results with the results of other authors as well as with the Galerkin method based on trigonometric polynomials and with a hybrid finite-volume finite-element method. We carry out a bifurcation analysis of the problem and indicate a scenario of passage from regular solutions to chaotic ones.

Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere

We study the bifurcation structure of zonal flows on a rotating sphere. The setting of our problem is similar to the Kolmogorov problem on a flat torus, where the vorticity forcing is given by a single eigenfunction of the Laplacian. First we prove the global stability of two-jet zonal flow for arbitrary Reynolds number and the rotation rate of the sphere. Then we study the bifurcation structure of steady solutions arising from three-jet zonal flow. In the non-rotating case, we find that two steady travelling-wave solutions bifurcate from a three-jet zonal flow via Hopf bifurcation. As the Reynolds number increases, steady-travelling solutions arise via pitchfork bifurcation from the steady-travelling solutions. On the other hand, in the rotating case, we find saddle-node bifurcations and closed-loop branches. We carry out time integration to study the properties of unsteady solutions at high Reynolds numbers. In the non-rotating case, the unsteady solution is chaotic and it wanders around the steady-travelling solutions bifurcating from three-jet zonal flow. We show that a linear combination of the steady and steady-travelling solutions gives a good approximation of the zonal-mean zonal flow of the unsteady solution, suggesting that the chaotic solution at high Reynolds numbers exists mostly within a relatively low-dimensional space spanned by the steady and steady-travelling solutions, which become unstable at low Reynolds numbers.

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers Mk1,Mk2,…,Mkd, 0≤k1<…<kd≤r, to guarantee the existence of an r-monotone function defined on the negative half-line R− and such that ‖x(ki)‖∞=Mki, i=1,2,…,d. We also discuss some applications of the obtained results and connections with other problems.

Linear Stability of Viscous Zonal Jet Flows on a Rotating Sphere

Linear stability of viscous zonal jet flows on a rotating sphere is investigated. This problem setting is similar to the Kolmogorov problem on the point that the basic solutions are expressed by a single eigenfunction of Laplacian of each manifold, a sphere and a torus. In the non-rotating case, as the number of jets increases the critical Reynolds number increases monotonically with the critical azimuthal wavenumber being \(m_{\text{c}}=2\). In the rotating cases, the zonal flows are stable for large rotation rates while the effect of small rotation does not always stabilize the zonal flows. We find that the critical rotation rate in the inviscid limit does not coincide with the inviscid one. This seeming contradiction between the inviscid limit stability and the inviscid stability is resolved by an observation that, as the Reynolds number increases, the growth rates of viscous unstable mode converge to zero in the regions where the inviscid zonal flow is stable while the viscous zonal flow is unstable.

The Landau--Kolmogorov inequality revisited

We consider the Landau--Kolmogorov problem on a finite interval which is to find an exact bound for $\|f^{(k)}\|$, for $0 < k < n$, given bounds $\|f\| \le 1$ and $\|f^{(n)}\| \le \sigma$, with $\|\cdot\|$ being the max-norm on $[-1,1]$. In 1972, Karlin conjectured that this bound is attained at the end-point of the interval by a certain Zolotarev polynomial or spline, and that was proved for a number of particular values of $n$ or $\sigma$. Here, we provide a complete proof of this conjecture in the polynomial case, i.e. for $0 \le \sigma \le \sigma_n := \|T_n^{(n)}\|$, where $T_n$ is the Chebyshev polynomial of degree $n$. In addition, we prove a certain Schur-type estimate which is of independent interest.

The Kolmogorov and Stechkin problems for classes of functions whose second derivative belongs to the Orlicz space

For any t ∈ [0, 1], we obtain the exact value of the modulus of continuity $\omega _N (D_t ,\delta ): = \sup \{ \left| {x'(t)} \right|:\left\| x \right\|_{L_\infty [0,1]} \leqslant \delta ,\left\| {x''} \right\|_{L_N^* [0,1]} \leqslant 1\} ,$, where L* N is the dual Orlicz space with Luxemburg norm and D t is the operator of differentition at the point t. As an application, we state necessary and sufficient conditions in the Kolmogorov problem for three numbers. Also we solve the Stechkin problem, i.e., the problem of approximating an unbounded operator of differentition D t by bounded linear operators for the class of functions x such that \(\left\| {x''} \right\|_{L_N^* [0,1]} \leqslant 1\).

Landau–Kolmogorov problem for a class of functions absolutely monotone on a finite interval

We solve the Landau–Kolmogorov problem for a class of functions absolutely monotone on a finite interval. For this class of functions, new exact additive inequalities of the Kolmogorov type are obtained.

Extendible bases and Kolmogorov problem on asymptotics of entropy and widths of some class of analytic functions

Let K be a compact set in an open set D on a Stein manifold Ω of dimension n. We denote by H ∞ D the Banach space of all bounded and analytic in D functions endowed with the uniform norm and by A K D a compact subset of the space CK consisted of all restrictions of functions from the unit ball 𝔹 H ∞ D . In 1950ies Kolmogorov posed a problem: does

Plane Kolmogorov flows and Takens–Bogdanov bifurcation without parameters: The singly reversible case

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we use the Kirchgassner reduction to obtain a spatial dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. In particular we describe the complete set B of all small bounded solutions. In the classical Kolmogorov case, B consists of periodic profiles, homoclinic pulses and a heteroclinic front-back pair. This is a consequence of the symmetry of the external force.