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## Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force

Discrete and Continuous Dynamical Systems. 2017. Vol. 37. No. 5. P. 2375–2393.

Bekmaganbetov K., Chechkin G., Chepyzhov V. V., Goritsky A.

We consider the 3D Navier--Stokes systems with randomly rapidly oscillating right--hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier--Stokes systems whose right--hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier--Stokes systems is uniquely solvable.

Danilov V., Gaydukov R., Mathematical notes 2015 Vol. 98 No. 4 P. 561–571

We consider the problem of a viscous incompressible fluid flow along a flat plate with a small solitary perturbation (of hump, step, or corner type) for large Reynolds numbers. We obtain an asymptotic solution in which the boundary layer has a double-deck structure. ...

Added: September 27, 2015

Gaydukov R., Danilov V., Наноструктуры. Математическая физика и моделирование 2016 Т. 15 № 1 С. 5–102

We study the existence conditions for a double-deck structure of a boundary layer in typical problems of incompressible fluid flow along surfaces with small irregularities (periodic or localized) for large Reynolds number. We obtain characteristic scales (a power of a small parameter included in a solution) which lead to the double-deck structure, and we obtain ...

Added: September 27, 2016

Stanislav Minkov, Ivan Shilin, / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2011.04824.

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows φ on S^2 for which the attractor of the Cartesian square of φ is smaller than the Cartesian square of the attractor of φ. In the example for the minimal attractors, the flow φ also has an SRB-measure such that its square does ...

Added: November 12, 2020

Stanislav Minkov, Shilin I., Qualitative Theory of Dynamical Systems 2021 Vol. 20 No. 3 Article 77

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows ϕ on S^2 for which the attractor of the Cartesian square of ϕ is smaller than the Cartesian square of the attractor of ϕ. In the example for the minimal attractors, the flow ϕ also has a global physical measure such that its ...

Added: September 16, 2021

Bekmaganbetov K., Chechkin G., Chepyzhov V. V., Communications on Pure and Applied Analysis 2020 Vol. 19 No. 5 P. 2419–2443

We consider reaction{diusion systems with random terms that oscillate rapidly in space variables. Under the assumption that the random functions are ergodic and statistically homogeneous we prove that the random
trajectory attractors of these systems tend to the deterministic trajectory attractors of the averaged reaction-diusion system whose terms are the average of the corresponding terms of the original system. Special ...

Added: November 11, 2020

Chepyzhov V. V., Bekmaganbetov K., Chechkin G., Applicable Analysis 2019 Vol. 98 No. 1-2 P. 256–271

We consider reaction–diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction–diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors ...

Added: November 11, 2020

Volk D., Ergodic Theory and Dynamical Systems 2014 Vol. 34 No. 2 P. 693–704

For a smooth manifold of any dimension greater than one, we present an open set of smooth endomorphisms such that any of them has a transitive attractor with a non-empty interior. These maps are m-fold non-branched coverings,m≥3. The construction applies to any manifold of the form S 1×M, where S 1 is the standard circle and Mis an arbitrary manifold. ...

Added: December 28, 2015

Gaydukov R. K., Borisov D. I., Mathematical notes 2016 Vol. 99 No. 5 P. 636–642

A fluid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure: a thin boundary layer (“lower deck”) and a classical Prandtl boundary layer (“upper deck”). The aim of this paper is to prove the existence and uniqueness of the ...

Added: May 18, 2016

Rabinowitch A. S., Journal of Mathematical Physics 2016 Vol. 57 P. 083103-1–083103-6

Added: October 5, 2019

Volk D., Kleptsyn V., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 339–365

A one-dimensional confined nonlinear random walk is a tuple of N diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for ...

Added: December 30, 2015

Солодовников Н. А., Труды Московского математического общества 2014 Т. 75 № 1 С. 15–24

We construct an open set of C2-diffeomorphisms which preserve the boundary of some manifold, and which have the following property: for each mapping, the basin of attraction of one component of the attractor is open and everywhere dense, but the basin of attraction of the second component is nowhere dense, though its measure is positive. ...

Added: November 29, 2015

Zelik S., Ильин А. А., Костянко А. Г., Математические заметки 2022 Т. 111 № 1 С. 54–66

We prove the existence of a global attractor of a regularized Euler–Bardina system with dissipation on the two-dimensional sphere and in arbitrary domains on the sphere. Explicit estimates for the fractal dimension of the attractor in terms of its physical parameters are obtained. ...

Added: May 9, 2023

Ilyashenko Y., Shilin I., Proceedings of the Steklov Institute of Mathematics 2012 Vol. 277 No. 1 P. 84–93

There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of ...

Added: February 5, 2013

Sinelshchikov D., Garashchuk I., Kudryashov N. A., Journal of Physics: Conference Series 2017 Vol. 788 P. 012013 -1–012013 -6

Non-linear dynamical systems describe many physical processes. In this work we investigate a three-dimensional Lorenz-like system - the Glukhovsky-Dolzhansky system. We consider analytical properties of the studied system. The problem of existence of meromorphic solution is discussed. We perform the Painlev`e test and find conditions imposed on parameters of the system for which meromorphic solutions ...

Added: December 16, 2019

Volk D., Discrete and Continuous Dynamical Systems 2014 Vol. 34 No. 5 P. 2307–2314

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite ...

Added: December 30, 2015

Rabinowitch A. S., Internarional Journal of Advanced Mathematical Sciences (UAE) 2013 Vol. 1 No. 4 P. 199–206

Added: October 5, 2019

Danilov V. G., Gaydukov R. K., Russian Journal of Mathematical Physics 2017 Vol. 24 No. 1 P. 1–18

The problem of flow of a viscous incompressible fluid in an axially symmetric pipe with small irregularities on the wall is considered. An asymptotic solution of the problem with the double-deck structure of the boundary layer and the unperturbed flow in the environment (the “core flow”) is obtained. The results of flow numerical simulation in ...

Added: September 28, 2016

Семенякин Н. С., Modern Physics Letters A 2014 Vol. 29 No. 35

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev{V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all ...

Added: April 4, 2018

Zlotnik A., Математическое моделирование 2012 Т. 24 № 4 С. 65–79

Дан краткий подход к построению баротропной квазигазодинамической (КГД) системы уравнений с помощью специальной регуляризации по времени баротропной системы уравнений Навье-Стокса вязкого сжимаемого газа. Этот подход обобщен на случай КГД системы с общими уравнениями состояния газа, массовой силой и источником тепла. Для баротропной КГД системы с потенциальной массовой силой выведено поточечное уравнение энергетического баланса. В качестве ...

Added: June 30, 2012

Условия разрешимости начальной задачи для систем нелинейных функционально-дифференциальных уравнений

Ларионов А. С., Симонов П. М., Sheina M. V., Вестник Тамбовского университета. Серия: Естественные и технические науки 2010 Т. 15 № 2 С. 542–549

Theorems of existence and uniqueness of Cauchy’s problem solution for systems of nonlinear functional and differential equations are proved. During the proof of the theorems the positivity of the Cauchy’s matrix corresponding linear system is used essentially. ...

Added: October 22, 2012

Rabinowitch A. S., International Journal of Advanced Mathematical Sciences (UAE) 2013 Vol. 1 No. 4 P. 199–206

Added: October 5, 2019

Zybin K., Sirota V. A., Physica Scripta 2013 Vol. 155

We analyze a particular simple case within the framework of the vortex filament (VF) model. It gives the same basic results and allows us to understand the general case better. Advantages and weaknesses of the simplification are considered. The introduction of stochastics into the Navier–Stokes equation, the evolution of VFs and longitudinal and transverse Euler ...

Added: October 20, 2014

Grines V., Pochinka O., Chilina E., / Cornell University. Series math "arxiv.org". 2023.

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set ...

Added: December 20, 2023

Rabinowitch A. S., Journal of Mathematical Physics 2014 Vol. 55 P. 093102-1–093102-11

Added: October 5, 2019