Anna Kostianko

Dr Anna Kostianko


Research Fellow
+44 (0)1483 689643
12 AA 04

Academic and research departments

Department of Mathematics.

My publications

Publications

Anna Kostianko, Edriss Titi, Sergey Zelik (2018)Large dispersion, averaging and attractors: three 1D paradigms, In: Nonlinearity31(12)R317 IOP Publishing

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-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.

An inertial manifold for the system of 1D reaction-diffusion-advection equations endowed by the Dirichlet boundary conditions is constructed. 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.

A Kostianko, S Zelik (2015)Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions., In: Communications on Pure and Applied Analysis14(5)pp. 2069-2094 AMER INST MATHEMATICAL SCIENCES

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.

Thomas J Bridges, Anna Kostianko, Sergey Zelik (2021)Validity of the hyperbolic Whitham modulation equations in Sobolev spaces, In: Journal of Differential Equations274pp. 971-995 Elsevier Inc

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.

Tom Bridges, Anna Kostianko, Guido Schneider (2020)A proof of validity for multiphase Whitham modulation theory, In: Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character.476(2243) The Royal Society Publishing

Proc. Roy. Soc. Lond. A (2020) 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 Schr\"odinger 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 Schr\"odinger equations and the multiphase Whitham modulation equations, is obtained.

A Kostianko, E Titi, S Zelik (2016)Large dispersion, averaging and attractors: three 1D paradigms, In: arXiv

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-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, adding the large dispersion term actually suppresses the non-trivial dynamics. According to the second scenario, neither the dissipation mechanism, nor the dynamics are essentially affected by the large dispersion and the limit dynamics remains complicated (chaotic). Finally, it is demonstrated in the third scenario that the dissipation mechanism is completely destroyed by the large dispersion, and that the diameter of the global attractor grows together with the growth of the dispersion parameter.

Anna Kostianko (2020)Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg-Landau equation, In: PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES ROYAL SOC

We present a new method of establishing the finitedimensionality of limit dynamics (in terms of bi- Lipschitz Mané 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.