Mathematical fluid dynamics laboratory
Grant Agreement No.: 14.Z50.31.0037
Project name: Investigation of mathematical problems of fluid dynamics
Name of the institution of higher learning: Voronezh State University
Fields of scientific research: Mathematics
The main goal of the project is to study a number of important and outstanding at this moment problems of mathematical fluid mechanics and obtaining a world-class scientific results. To achieve this goal we have picked up the team the core of which is made up of world-class scientists. It is supposed to create a laboratory in which the team will carry out scientific research.
The second major objective of this project is to involve young scientists and students in scientific research to give them the opportunity to consolidate in science. For this it is supposed to create by leading scientists special courses of lectures for students of the Faculty of Mathematics and the Department of Applied Mathematics and Mechanics, and to carry out seminars.
In connection with the set goals of the project it is possible to emphasize the following objectives (scientific problems) of the project:
1. Investigation of the in/out flow problem for a viscous compressible gas equations and its application to the mathematical modeling of the work of thermal machines and heat engines.
2. Analysis of variational problems of conformal geometry and potential theory and application of these results to the existence of global smooth periodic solution to the problem of nonlinear waves problem in the deep ocean covered by ice.
3. The proof of an analog of Liouville Theorem for the steady system of Navier-Stokes equations in the 3D case with axial symmetry under assumption that velocity tends to zero at infinity.
4. The development of the theory of global and trajectory attractors for dissipative infinite-dimensional dynamical systems and the application of this theory to study the long-term and limit behavior of solutions of some fundamental models arising in mathematical physics, fluid dynamics and geophysical fluid dynamics, which are described by nonlinear partial differential equations.
5. Investigation of the solvability of a lot of initial value problems of non-Newtonian fluid dynamics (models of fluid with memory, Bingham model, etc.) and the study of their attractors in both autonomous and non-autonomous cases (uniform attractors and pullback-attractors).
6. Investigation of solvability, optimal control and qualitative behavior of alpha-models of fluid mechanics.
7. The construction of stabilization theory for the system of normal type, connected with Helmholtz system by means of starting control and to construct similar theory when impulse or distributed controls supported on prescribed subdomain of the 3D torus are used. Using this result to construct the theory of nonlocal stabilization for 3D Helmholtz system by impulse as well as by distributed controls.
8. Investigation of the problem of small motion of hyperbolic type viscoelastic body. The study of uniform exponential stability of an abstract strongly continuous semigroup generated by the operator unit of a special kind.
9. The next goal is to develop the theory of the unbalanced optimal transport and to reveal its relations with other areas of mathematics such as metric geometry, differential geometry, calculus of variations, Hamilton-Jacobi equations, convex analysis, gradient flows on metric spaces, evolutionary PDEs, dynamical systems.
Also in connection with the second goal of the project it is necessary to highlight the problem of creating workplaces for young scientists and students of Voronezh State University in the laboratory, which will be created for the project.
Plotnikov Pavel Igorevich
Date of Birth: 04.11.1947
Academic degree and title: Dr.Sci.
Job Title: Novosibirsk State University
Field of scientific interests: Partial differential equations theory, hydrodynamics, nonlinear waves theory, KAM theory, nonlinear functional analysis
P.I. Plotnikov - specialist in the theory of differential equations and mathematical physics. His scientific activity is connected with mathematical problems of nonlinear wave theory, mathematical questions of the theory of phase transitions and the theory of materials, as well as the theory of viscous gas dynamics. His results on the Stokes problem in the theory of waves and the problem of small divisors in the theory of Hamiltonian systems with an infinite number degrees of freedom gain recognition among scientific community. Among the results obtained in this direction are the proof of the first and the second proof Stokes conjectures contained in 1880 Stokes famous work of extreme waves, the proof of the solvability of the standing wave, the proof the existence of an infinite number of secondary bifurcations of solutions to the problem of solitary waves, the proof of the existence of three-dimensional asymmetric wave packets that propagate along the surface water at a constant speed. In order to solve these problems there
were developed: the method of analytic extension of solutions of free boundary problems, the version of the theory of Nash-Moser for the problem of standing waves, the infinite-dimensional version of the method Floquet for pseudo-differential equations with small divisors, the theory of
Conley topological index for the critical points of smooth functional in the infinite-dimensional spaces.
In 1994, the works by P.I. Plotnikov on nonlinear wave theory were awarded Lavrentyev Prize the Russian Academy of Sciences.