Searcharxiv⌕ Search

arXiv subjects

Anna Kostianko

Publications and source records attributed to Anna Kostianko.

At least 19 recordsLinked to original sources
arXiv2025

Multi-vortices and lower bounds on the attractor dimensions for 2D Navier--Stokes equations

We present a principally new method for obtaining the lower bounds for the attractors dimensions of the equations related with hydrodynamics, which is not based on the Kolmogorov flows, and apply it to the classical 2D Navier--Stokes equations in a bounded domain as well as for the Navier--Stokes equations with Ekman damping inthe whole plane. In particular, in the case of bounded domains, we give the lower bounds, which are similar to the well-known estimate on a torus. In both cases our estimates are sharp. Note that no lower bounds for these two cases were known before. \par We suggest to use the so-called multi-vortex, which consists of a well-separated Vishik vortices (i.e., spectrally unstable localized in space flows constructed by M.M. Vishik), as the analogue of the Kolmogorov flows. Note also that this method reproduces the known result on the torus and that it is applicable to many other equations of hydrodynamics.

math.AP↗
arXiv2023

Non-concentration phenomenon for one dimensional reaction-diffusion systems with mass dissipation

Reaction-diffusion systems with mass dissipation are known to possess blow-up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension one, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, i.e. nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this article, we remove this extra entropy assumption completely and obtain global boundedness for reaction-diffusion systems with cubic intermediate sum condition. The novel idea is to show a non-concentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space $\mathsf{M}^{1,δ}(Ω)$ for some $δ>0$. As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction-diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov-Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly-super cubic intermediate sum condition.

math.AP↗
arXiv2022

Attractors for semigroups with multi-dimensional time and PDEs in unbounded domains

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semi-invariant with respect to some direction. In particular, the theory is applicable to the exterior domains.

math.AP↗
arXiv2022

Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces

We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damped wave equation with Dirichlet boundary conditions, quintic 3D wave equation with periodic boundary conditions and 2D Navier-Stokes system in a bounded domain. The examples of finite-dimensional uniform attractors for some special external forces which are not translation-compact are also given.

math.AP↗
arXiv2022

Applications of the Lieb--Thirring and other bounds for orthonormal systems in mathematical hydrodynamics

We discuss the estimates for the $L^p$-norms of systems of functions that are orthonormal in $L^2$ and $H^1$, respectively, and their essential role in deriving good or even optimal bounds for the dimension of global attractors for the classical Navier--Stokes equations and for a class of $α$-models approximating them. New applications to interpolation inequalities on the 2D torus are also given.

math.AP↗
arXiv2021

Trajectory attractors for 3D damped Euler equations and their approximation

We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ endowed with periodic boundary conditions as well as their damped Euler limit $α\to0$. We prove that despite the possible non-uniqueness of solutions of the limit Euler system and even the non-existence of such solutions in the distributional sense, the limit dynamics of the corresponding dissipative solutions introduced by P.\,Lions can be described in terms of attractors of the properly constructed trajectory dynamical system. Moreover, the convergence of the attractors $\Cal A(α)$ of the regularized system to the limit trajectory attractor $\Cal A(0)$ as $α\to0$ is also established in terms of the upper semicontinuity in the properly defined functional space.

math.AP↗
arXiv2021

Determining functionals and finite-dimensional reduction for dissipative PDEs revisited

We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common paradigm, it is shown that the optimal number of determining functionals (the so-called determining dimension) is strongly related to the proper dimension of the set of equilibria of the considered dynamical system rather than to the dimensions of the global attractors and the complexity of the dynamics on it. In particular, in the generic case where the set of equilibria is finite, the determining dimension equals to one (in complete agreement with the Takens delayed embedding theorem) no matter how complex the underlying dynamics is. The obtained results are illustrated by a number of explicit examples.

math.AP↗
arXiv2021

Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions

We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the recently suggested method of spatio-temporal averaging.

math.AP↗
arXiv2021

Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations

The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $α\to0$ and $γ\to0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.

math.AP↗
arXiv2021

Smooth extensions for inertial manifolds of semilinear parabolic equations

The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the $C^{1,\varepsilon}$-smooth IM of a minimal dimension). The proof is strongly based on the Whitney extension theorem.

math.AP↗
arXiv2020

Validity of the hyperbolic Whitham modulation equations in Sobolev spaces

It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrödinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.

math.AP↗
arXiv2020

A proof of validity for multiphase Whitham modulation theory

It is proved that approximations which are obtained as solutions of the multiphase Whitham modulation equations stay close to solutions of the original equation on a natural time scale. The class of nonlinear wave equations chosen for the starting point is coupled nonlinear Schrödinger equations. These equations are not in general integrable, but they have an explicit family of multiphase wavetrains that generate multiphase Whitham equations which may be elliptic,hyperbolic, or of mixed type. Due to the change of type, the function space setup is based on Gevrey spaces with initial data analytic in a strip in the complex plane. In these spaces a Cauchy- Kowalevskaya-like existence and uniqueness theorem is proved. Building on this theorem and higher-order approximations to Whitham theory, a rigorous comparison of solutions, of the coupled nonlinear Schrödinger equations and the multiphase Whitham modulation equations, is obtained.

math.AP↗
arXiv2020

Inertial manifolds via spatial averaging revisited

The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by G. Sell and J. Mallet-Paret. We present a universal approach which covers the most part of known results obtained via this method as well as gives a number of new ones. Among our applications are reaction-diffusion equations, various types of generalized Cahn-Hilliard equations, including fractional and 6th order Cahn-Hilliard equations and several classes of modified Navier-Stokes equations including the Leray-$α$ regularization, hyperviscous regularization and their combinations. All of the results are obtained in 3D case with periodic boundary conditions.

math.AP↗
arXiv2020

Reaction-diffusion systems with supercritical nonlinearities revisited

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

math.AP↗
arXiv2020

Bi-Lipschitz Mane projectors and finite-dimensional reduction for complex Ginzburg-Landau equation

We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mane projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of 3D complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet-Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.

math.AP↗
arXiv2019

Kwak Transform and Inertial Manifolds revisited

The paper gives sharp spectral gap conditions for existence of inertial manifolds for abstract semilinear parabolic equations with non-self-adjoint leading part. Main attention is paid to the case where this leading part have Jordan cells which appear after applying the so-called Kwak transform to various important equations such as 2D Navier-Stokes equations, reaction-diffusion-advection systems, etc. The different forms of Kwak transforms and relations between them are also discussed.

math.AP↗
arXiv2017

Inertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.

math.AP↗