17 citations · 39 across the 18 of their papers we have counts for
22 papers
Multi-vortices and lower bounds on the attractor dimensions for 2D Navier--Stokes equations
Anna Kostianko, Alexei Ilyin, Dominic Stone +1
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 f…
Attractors for semigroups with multi-dimensional time and PDEs in unbounded domains
Anna Kostianko, Sergey Zelik
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…
Non-concentration phenomenon for one dimensional reaction-diffusion systems with mass dissipation
Juan Yang, Anna Kostianko, Chunyou Sun +2
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 o…
Entropy estimates for uniform attractors of dissipative PDEs with non translation-compact external forces
Yangmin Xiong, Anna Kostianko, Chunyou Sun +1
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-c…
Applications of the Lieb--Thirring and other bounds for orthonormal systems in mathematical hydrodynamics
Alexei Ilyin, Anna Kostianko, Sergey Zelik
We discuss the estimates for the -norms of systems of functions that are orthonormal in and , respectively, and their essential role in deriving good or even optima…
Trajectory attractors for 3D damped Euler equations and their approximation
Alexei Ilyin, Anna Kostianko, Sergey Zelik
We study the global attractors for the damped 3D Euler--Bardina equations with the regularization parameter and Ekman damping coefficient endowed with periodic boundary…
Determining functionals and finite-dimensional reduction for dissipative PDEs revisited
Varga Kalantarov, Anna Kostianko, Sergey Zelik
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 th…
Dimension estimates for the attractor of the regularized damped Euler equations on the sphere
Alexei Ilyin, Anna Kostianko, Sergey Zelik
We prove existence of the global attractor of the damped and driven Euler--Bardina equations on the 2D sphere and on arbitrary domains on the sphere and give explicit estimates of…
Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions
Anna Kostianko, Chunyou Sun, Sergey Zelik
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…
Sharp upper and lower bounds of the attractor dimension for 3D damped Euler-Bardina equations
Alexei Ilyin, Anna Kostianko, Sergey Zelik
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter and Ekman damping coefficient is…
Smooth extensions for inertial manifolds of semilinear parabolic equations
Anna Kostianko, Sergey Zelik
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…
Inertial manifolds via spatial averaging revisited
Anna Kostianko, Xinhua Li, Chunyou Sun +1
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. M…
Reaction-diffusion systems with supercritical nonlinearities revisited
Anna Kostianko, Chunyou Sun, Sergey Zelik
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 s…
Validity of the hyperbolic Whitham modulation equations in Sobolev spaces
Thomas J. Bridges, Anna Kostianko, Sergey Zelik
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 modulatio…
A proof of validity for multiphase Whitham modulation theory
Tom Bridges, Anna Kostianko, Guido Schneider
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 s…
Bi-Lipschitz Mane projectors and finite-dimensional reduction for complex Ginzburg-Landau equation
Anna Kostianko
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…
Kwak Transform and Inertial Manifolds revisited
Anna Kostianko, Sergey Zelik
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…
Inertial manifolds for 1D reaction-diffusion-advection systems. Part II: periodic boundary conditions
Anna Kostianko, Sergey Zelik
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in \cite{KZI} and it is devoted to the case of per…
Inertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions
Anna Kostianko, Sergey Zelik
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…
Large dispersion, averaging and attractors: three 1D paradigms
Anna Kostianko, Edriss Titi, Sergey Zelik
The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg…
Inertial manifolds for the 3D modified-Leray- model with periodic boundary conditions
Anna Kostianko
The existence of an inertial manifold for the modified Leray- model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averagi…
Inertial Manifolds for the 3D Cahn-Hilliard Equations with Periodic Boundary Conditions
Anna Kostianko, Sergey Zelik
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging…