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Laboratory for Mathematical Hydrodynamics

Invited researcher Pavel Igorevich Plotnikov
Contract number
14.Z50.31.0037
Time span of the project
2017-2019

As of 15.02.2021

9
Number of staff members
71
scientific publications
General information

Name of the project: Research of mathematical hydrodynamics problems

Strategy for Scientific and Technological Development Priority Level: е


Goals and objectives

Research directions: Mathematical hydrodynamics

Project objective: Research of a variety of important and currently unsolved problems of mathematical hydrodynamics; engaging young scientists, postgraduates and undergraduates in research to give them an opportunity to strengthen their involvement in science


The practical value of the study

  • There are obtained theorems on the existence of solutions to the problem of nonlinear steady-state waves on the surface of an ice-covered ocean. An analog of Liouville's theorem for axisymmetric flows of a viscous fluid (stationary Navier-Stokes equations) with rotation is obtained. The general construction of trajectory and global attractors of evolutionary equations with memory is described. Existence theorems for a minimal trajectory pullback-attractor and a global pullback-attractor are proved both for weak solutions of a non-autonomous medium with memory and weak solutions of the model of a Bingham medium motion in the non-autonomous case. Existence theorems for weak solutions are proved for the Leray, Navier–Stokes, and Voigt-alpha models with viscosity coefficients depending on temperature.

  • In 2018 and 2019, field measurements of internal waves were carried out at the Marine Experimental Station "Mys Shulstza" of the V.I. Il'ichev Pacific Oceanological Institute Far Eastern Branch Russian Academy of Sciences, and experimental data were obtained. Numerical modeling of nonlinear internal waves was carried out and a model of nonlinear internal waves (including Kelvin and Poincaré) was built. A comparative analysis of the obtained experimental data with the results of numerical simulation was carried out. The structure of solutions and the solvability of a boundary value problem describing nonlinear waves in a continuously stratified fluid over an obstacle were investigated. The interaction of coupled traveling waves is studied on the base of the model of weakly coupled Kortweg- de Vries equations. Solutions of many inverse problems on the reproduction of the structure of nonlinear packets of internal waves are constructed. The temperature field and the boundaries of the layers are reconstructed during the passage of a near-surface solitary wave and wave boron. The constructed model was verified by comparing it with experimental data obtained not only at the Marine Experimental Station "Mys Shulstza" of the V.I. Il'ichev Pacific Oceanological Institute Far Eastern Branch Russian Academy of Sciences, but also with experimental data obtained in the South China Sea (Lien, Henyey, Ma and Yang, 2004).

Education and career development:

  • In 2017, five employees of the Laboratory of Mathematical Hydrodynamics have completed career enhancement programs on the topic: "Partial Differential Equations and Their Applications to Mathematical Hydrodynamics" at the Faculty of Mathematics of the Middle East University, Nicosia, Turkish Republic of Northern Cyprus.

  • In 2018, five laboratory employees have completed career enhancement programs at the International Center for Mathematics (Portugal).

  • In 2019, five employees of the laboratory of mathematical hydrodynamics have completed career enhancement programs on the topic: “Multiphase non-Newtonian fluids: mathematical modeling and calculations” at the IUSTI laboratory in Marseille, France.

  • The international scientific conference "Modern methods and problems of mathematical hydrodynamics" in 2017, "Modern methods and problems of mathematical hydrodynamics-2018" in 2018, and "Modern methods and problems of mathematical hydrodynamics-2019" in 2019 were held.

  • A scientific seminar "Mathematical models of shallow water shear flows" was held with a visiting lecturer from the University of Marseille.

  • 1 candidate dissertations, 5 master dissertations, and 5 bachelor dissertations have been defended. Four of the Laboratory employees have been admitted to postgraduate school.

  • Six lecture courses were created and introduced into the educational process of the Faculty of Mathematics of Voronezh State University: "Applications of the theory of differential equations to geometry", "Navier-Stokes equations of a compressible fluid", "Applications of differential inclusions to problems of optimal control", "Mathematical models of movement of polymer solutions", "Alpha-models of the equations of hydrodynamics", and "Approximation-topological method for the solvability of the equations of the hydrodynamics of viscoelastic media".

Other results: Employees of the Laboratory participated in 18 international conferences, scientific schools and delivered 51 keynotes.

Collaborations:

V.I. Il'ichev Pacific Oceanological Institute Far Eastern Branch Russian Academy of Sciences: an agreement on scientific cooperation was signed, a number of joint studies were carried out, based on the results of which 2 articles were published..

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Fursikov A., Osipova L.
On the nonlocal stabilization by starting control of the normal equation generated from Helmholtz system // Science Chine Mathematics. – 2018. – Vol. 61. – Issue 11. – pp. 2017-2032.
Plotnikov P.I., Toland J.F.
Variational Problems in the Theory of Hydroelastic Waves // Philosophical transactions of the Royal society A-mathematical physical and engineering sciences. – 2018. – Vol. 376. – Issue 2129 – Article ID:20170343.
Zvyagin V.G., Orlov V.P.
Solvability of one non-Newtonian fluid dynamics model with memory // Nonlinear Analysis. – 2018. – Vol. 172. – pp. 73–98.
Zvyagin A.V.
Attractors for model of polymer solutions motion // Discrete And Continuous Dynamical Systems. – 2018. – Vol. 38. – № 12. – pp. 6305–6325.
Seregin G.A., Shilkin T.N
Liouville-type theorems for the Navier-Stokes equations // Russian Mathematical Surveys. – 2018. – Vol. 73. – Issue 4. – pp. 661-724.
Korobkov M.V., Pileckas K., Russo R.
On Convergence of Arbitrary D-Solution of Steady Navier-Stokes System in 2D Exterior Domains // Archive for Rational Mechanics and Analysis. – 2019. – Vol. 233. – Issue 1. – pp. 358-407.
Zvyagin A.V.
Weak solvability and convergence of solutions for the fractional Voigt-α model of a viscoelastic medium // Russian Mathematical Surveys. – 2019. – Vol. 74. – № 3. – pp. 549-551.
Plotnikov P.I., Soko lowski J.
Boundary Control of the Motion of a Heavy Piston in Viscous Gas // SIAM Journal on Control and Optimization. – 2019. – Vol. 57. – Issue 5. – pp. 3166–3192.
Kornev S., Obukhovskii V., Yao J.-C.
Random integral guiding functions in the periodic problem for random differential inclusions with causal multioperators // Journal of Differential Equations. – 2020. - Volume 268, Issue 10. - pp. 5792-5810.
Fursikov A., Osipova L.
On the nonlocal stabilization by starting control of the normal equation generated from Helmholtz system // Science Chine Mathematics. – 2018. – Vol. 61. – Issue 11. – pp. 2017-2032.
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