Embedded Files
I conduct research in the area of applied dynamical systems in a broad sense: fluid dynamics, traveling waves, pattern formation.
More specific projects are:
Viscous fingering phenomenon in fluid dynamics
Shadowing in partial differential equations
Spatially continuous and spatially discrete reaction-diffusion equations
Hysteresis
Recently I got interested in quantum computing.
Fluid dynamics and applications.
I study fluid dynamics in porous media motivated by applications in oil and gas industry. Injection of less viscous fluid to a more viscous one generates instabilities, which are called “viscous fingers”. This phenomenon has a negative impact on various flooding schemes in oil fields. The model is a system of PDEs consisting of conservation of mass, incompressibility condition and Darcy's law. The goal is to study the size of the mixing zone, where instabilities are located. The project involves construction and analysis of simplified models of various nature: non-local PDEs, cascade of travelling waves.
Talk 06.2025: Youtube
Pseudotrajectories and shadowing.
I investigate how similar the properties of pseudotrajectories are to those of exact trajectories (shadowing). I showed that the properties of infinite pseudotrajectories and exact trajectories are very alike only for uniformly hyperbolic systems. The aim of this research is to establish similarities between pseudotrajectories and exact trajectories for a broad class of not hyperbolic systems. I consider dynamical systems with discrete and continuous time, as well as group actions.
Mathematics of Quantum Computing.
I study the model of quantum computing based on tensor products of complex linear spaces. It allows to construct algorithms independently of physical realization. The main difficulty is that current implementation of the devices is very noisy. I plan to do research in two directions:
(1) solving linear equations on quantum annealers;
(2) sampling from Gaussian processes using quantum computers.
Video lectures 2025: Youtube Playlist Material of the course
Reaction-diffusion equations with hysteresis.
I study mathematical models of various biological phenomena with a hysteresis operator with memory. We explore a new mechanism for pattern formation, which we call ``rattling''. We aim at obtaining a detailed description of the phenomenon and proving that it persists under most approximations such as spatial discretisation, regularizations by slow-fast and delay. It seems the phenomena is related to the evolution of the Young measures. Results of this project allow to describe mechanism of pattern formation for a wide class of biological systems from bacteria growth in a petri plate to pulse propagation in neural networks on local (rattling subsection) and global (transverse subsection) scales.
Shadowing in PDE.
The goal of this project is to study shadowing in infinite-dimensional dynamical systems. Currently I consider two subprojects:
Aposteriory estimates for numerical simulations of parabolic problems;
Rigorous comparison of miscible displacement determined by Darcy law and Transverse flow equilibrium model.
Depinning bifurcation.
I study differences between spatially discrete and spatially continuous reaction-diffusion equations. As a first goal I see the description of the phase transition for travelling waves, which is called (de-)pinning. The behaviour of such systems is well investigated far from the transition point. Surprisingly not much is known for parameter values near the phase transition. The main obstacle is that the system poses in a symmetry under action of a non-compact group, which makes the bifurcation analysis difficult.
Page updated
Google Sites
Report abuse