
Research Interests:
 Surfactant driven thin film flows and biomedical applications.
 PDE Analysis: thin film equations, spectral stability problems in fluid dynamics.
 Numerical Analysis: Scientific computations in PDEs
 Applied Operator Theory: spectral theory of Jselfadjoint operators.
 SturmLiouville problems with spectral parameter dependent boundary conditions.
Undergraduate students research projects
Brief summary of research projects:
2011– ...
Research project 1: ”Classes of uniqueness for solutions of the thin film equations. Surfactant driven thin film flows.”
Applications: surfactant replacement therapy and surfactant based drug delivery systems. The surfactant, produced by our alveolar cells, changes the surface tension of the lung, lowering the normal air–water surface tension of approximately 70 mJ/m^2 to
values below 5 mJ/m^2. By lowering the alveolar surface tension, the energy required to inflate the lungs during inspiration is minimized, preventing lung collapse during exhaling.
Research in progress: uniqueness for waiting time solutions, local and global in time existence and asymptotic behaviour of nonnegative weak solutions to a coupled system of
degenerate parabolic equations, finite speed and waiting time phenomena.
Collaboration: R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine and University of Nottingham ), John R. King ( University of Nottingham)
Research project 2: ”Finite–time blow up and long–wave unstable thin film equations.”
Applications: such equations arise in the modelling of fluids and materials. For
example, the equation with n=m=1 describes a thin jet in a HeleShaw cell.
Research in progress: shorttime existence, longtime existence, finite speed of propagation, and finitetime blowup of nonnegative solutions for
longwave unstable thin film equations.
Collaboration: R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine and University of Nottingham ), M. Pugh (University of Toronto)
2009– 2010
Research project 3: ”Analytical and Numerical Analysis of Thin Film Equations With Convection.”
Application: Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include coating flows, lava flows, dynamics of continental ice sheets, tearfilm rupture, and surfactant replacement therapy.
Analytical results: existence of nonnegtaive periodic weak solutions was proved for some reasonable restrictions on initial data.
Collaboration: M. Pugh (University of Toronto), R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine)
Analytical results: the lower bound on the convergence rate of nonnegative weak solution to the steady state.
Collaboration: A. Burchard (University of Toronto), B. Stephens (University of Washington)
Analytical results: analysis of stability of periodic, solitary, and shock waves in liquid films on vibrating substrates.
Collaboration: E. Benilov (University of Limerick)
2007 – 2008
Research project 4: ” Nature of IllPosedness of ForwardBackward Heat Equation ”
Application: dynamic of a thin film of liquid which is entrained on the inside of a rotating cylinder for example coating of fluorescent light bulbs or distribution of liquid thermosetting plastic in the rotating mould.
Analytical results: completness with the absence of the Riesz basis property was proved for the system of eigenfunctions of the indefinite convectiondiffusion operator.
Collaboration: I. Karabash (University of Calgary, Canada), S.G. Pyatkov (Sobolev Institute of Mathematics, Novosibirsk, Russia), V. Strauss (Universidad Simón Bolívar, Venezuela).
Research project 5: “Eigenvalue Distributions of Quadratic Matrix Pencils with Symmetries”
Application: stability of vortices in multidimensional discrete lattices.
Analytical results: upper bound was found for the number of spectrally unstable eigenvalues.
Collaboration: D. Pelinovsky (McMaster University, Canada).
2003 – 2007
Research project 6: “Spectral Stability of Solitary Waves in Dynamical Systems” (Doctoral Thesis, McMaster University)
Application: dynamic of the magnetoacoustic waves in plasma and capillarygravity water waves, stability of gap solitons in trapped BoseEinstein condensates, stability of vacuum.
Analytical results: Criteria for stability and instability of solitons were derived in terms of dimensions of signdefinite invariant subspaces using Pontryagin space decomposition method. By one of these criteria it was proved that embedded eigenvalues of negative Krein signature are structurally stable in a linearized fifth – order KdV equation.
Numerical results: the numerical method was constructed to find twopulse solutions for the fifthorder KdV equation; the MPI (parallel programming library) was used for the implementation of the code
Collaboration: T. Azizov (Voronezh State University, Russia), A. Comech
(Texas A&M University, US), Mason A. Porter (California Institute of Technology)
Advisor: Dr. D. Pelinovsky (McMaster University, Canada).
1994 – 2002
Research project 7: “Generalized Resolvent Method and SturmLiouville Problem with Spectral Parameter Dependent Boundary Conditions”.
Application: transverse vibrations of rotating beam with tip mass and vibrations of string with set of discrete loads.
Analytical results: the asymptotic of eigenvalues was obtained for different type of SturmLiouville problems where the familiar DirichletNeumann boundary conditions were replaced by boundary conditions that depend on the spectral parameter; the inverse problem was solved for the Nevanlinna type boundary conditions
Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia).
1990 – 1993
Research project 8: “Boundary Operators and Entire Extensions of Symmetric Operators”.
Analytical results: entire extension of a secondorder differentiation operator was constructed using the boundary operator method.
Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia)
1988 – 1990
Research project 9: “Automated Construction of Different Sectional Views for a Threedimensional Model of the Russian Space Station ‘Mir’“. (Master Thesis, Moscow Institute of Physics and Technology)
Numerical results: the numerical algorithm of calculating the area of the shadow of the complicated threedimensional object was constructed and implemented as a Pascal programming language code; the surfaces of objects (space modules of the station “Mir”) were approximated using triangulation; the dynamical tree – structure was used for a database of the approximated objects.
Advisor: Dr. M. Komarov (M.V.Keldysh Institute of Applied Mathematics, Russia)
 