Laboratory for Mathematical Hydrodynamics

• A proof of the theorem on the existence of a weak solution of the free boundary problem for equations of viscous gas dynamics modeling the motion of a heavy valve in a cylinder under the influence of the pressure of the viscous gas.

• A proof of the global theorem on the existence and uniqueness of a strong solution of one-dimensional initial-boundary value problem of the motion of multi-speed mixtures of viscous compressible fluids with a nondiagonal viscosity matrix in a bounded region with conditions allowing for  sticking of its components on the boundary.

• A proof of the theorem on the existence and regularity of solutions in the problem of joint minimization of the Willmore and Dirichlet functionals. We have proven the existence of three-dimensional, hydroelastic, nonlinear periodic waves propagating along a pool of infinite depth.

• A proof of the Bernoulli's law for axis-symmetricl Sobolev solutions of the Euler equation in unbounded domains

• A proof of the statement on the convergence of the trajectory attractor of a dissipative 2D Navier-Stokes system in the limit at a viscosity tending to zero to the trajectory  atractor of the corresponding dissipative 2D Euler system a strong Hausdorff metric.

• A proof of the theorems of the existence of weak solutions of the fractional Voigt viscoelasticity model with memory along trajectories of motion and the initial-boundary value problem for a system of hydrodynamics with memory that is a fractional   counterpart of the Voigt viscoelasticity model. A proof of the existence of a weak solution  of an initial-boundary value problem that describes the dynamics of a viscoelastic medium with memory on an infinite interval. A proof of the existence of a  unique solution of the initial-boundary value problem for systems of hydrodynamics with memory described by degenerate parabolic and elliptical equations. A proof of  the theorem on the existence of weak solutions of models of motion of a Bingham fluid. A proof of the theorems of the existence of attractors of solutions of autonomous systems of hydrodynamics with memory and attractors of a model of motion  of a Bingham fluid.

• A proof of the theorems of the existence of weak and dissipative solutions (we have also indicated a relation of strong and dissipative solutions, alpha models of motion  of fluid with memory). A proof of the theorem on the existence of weak solutions  of the Pavlovsky-Oskolkov model. A proof of the theorem on the existence of dissipative solutions (we have indicated the relation between strong and dissipative solutions) of the initial-boundary value problem for the Jeffreys-Oldroyd alpha model. A proof of the theorem on the existence of weak solutions of the alpha-model of the motion of solutions  polymers.

• A description of the built theory of the control of a system of normal type related to the Helmholtz equation via start and impulse control. A description of the developed nonlocal method of stabilization of three-dimensional Helmholtz and Navier-Stokes systems by impulse control.

• A proof of the theorem on the uniform exponential stability of a strongly continuous semigroup of bounded operators generated by an operator block of a special type. The results of solving two problems of the mechanics of continuous media (problems of the motion of a viscoelastic body and the problem of minor motions of a viscoelastic compressible Maxwell fluid filling a bounded uniformly rotating region) with the use of the specified theorem.

• A description of the strict theory of gradient flows and  geodesic convexity with respect to the geometrical structure related to this metric. A description of the De Giorgi-Jordan-Kinderlehrer-Otto variational scheme for building solutions of specified gradient flows.

• A description of the developed theory of internal Poincaré and Kelvin waves in a region with variable depth and a curvilinear boundary. Results of an analysis of the possibilities of generation of internal waves. A description of the developed models and computer tools for solving problems of generation and propagation of internal Poincaré and Kelvin waves based on nonlinear shallow water equations for laminar flow of fluid with mass exchange.

• The construction of a hierarchy mathematical models describing the wave dynamics of an oceanic environment accounting for the features of topography, dispersion and breaking of internal waves. Conducting an analysis and systematization of field data collected in various regions of the World Ocean. For this purpose we used  the latest data on deep water currents in the Atlantic (the Vema Channel, the Chain and Romanche fracture zones, the Discovery Passage) as well as on the transformation of internal waves in the shell zone of the Sea of Japan and the South China Sea. A  description of the developed numerical and analytical model of internal waves in the ocean accounting for the regional features of field experimental data. We have developed a mathematical model of the Green-Naghdi and Korteweg-de Vries type for describing strongly nonlinear internal waves factoring in the thin stratification of the density field and the amplitude dispersion. A systematization of the features of the formation of internal waves depending on the region. Obtaining asymptotic large-time asymptotic solutions for nonlinear equations    describing the propagation of nonstationary internal waves in a stratified fluid caused by  the motion of a submerged body under ice cover.

• A proof of the global solvability of the problem of optimization of the operation of a Stirling engine.

• A proof of the existence of bi-Lipschitz conformal coordinates on a surface that is an extremal of the Dirichlet and Willmore functionals. A proof of the statement that the set of features of such a parametrization has a  zero Hausdorff measure in any positive dimensionality.

• A proof of an analog of the Liouville's theorem for axisymmetric flows  of an ideal fluid (Euler equations) without rotation. As a consequence, we have obtained a proof of an analog of the the Liouville's theorem for axisymmetric flows of the Navier-Stokes system without rotation.

• A proof of the statement that if random functions are ergodic and statistically homogeneous in spatial variables or in time, then the trajectory attractors of  the 3D Navier-Stokes system converge in a weak topology to the trajectory attractor of the average 3D Navier-Stokes system whose deterministic external force is obtained by averaging the external forces of the initial random 3D  Navier-Stokes systems.

• A proof of the theorem on the existence of uniform attractors of solutions of  a non-autonomous system of hydrodynamics with memory and a model of the motion of the Bingham medium.

• A proof of the theorem on the existence of optimal control with feedback for a number of alpha-models of hydrodynamics. Particularly, a proof of the existence of optimal control with feedback for the alpha-Leray model, the alpha-Navier-Stokes model, the Pablovsky system of equations.

• A description of  a nonlocal method of stabilization of systems of normal type as well as 3D Helmholtz and Navier-Stokes systems by distributed control. Results of a resurch of the interaction of the normal and the tangential operators.

• A proof of the theorem on the strong solvability of initial-boundary value problems  describing minor motions of a viscoelastic body of hyperbolic type. A proof of the statement on the stability and stabilization of the solution of the researched problem.

• A description of the built theory and results of an analysis of the metric, geometric and topological properties of the MCF distance on the space of finite gradient vector Radon measures and on the space of bounded variation functions. A description of a new  model for image  reconstruction relying on MCF distance. A proof of the existence of  a solution of this model of image recognition, the correctness and the qualitative properties.

• Results of numerical  computations of the propagation of nonlinear internal waves (including Poincaré and Kelvin waves) for specific hydrodynamic systems, in particular, for the coastal zone of the sea. Results of field observations and measurements of the hydrodynamic parameters of flows in specific water reservoirs. An analysis of field observation data and their comparison with the results of the numerical modeling.

• Results of a comparative analysis of the results of a modeling of the wave dynamics of the oceanic environment and field data for specific areas of the World Ocean. A description of the developed hydrodynamic model of a specific region of the ocean accounting for the features of the wave dynamics related to the passage of nonlinear  internal waves. A description of a mathematical model for determining the wave field  with passage of a bundle of internal nonlinear waves and an analysis of the results of a numerical modeling.

• Proofs of the theorems on the existence of weak solutions of initial-boundary value problems for models of a viscoelastic fluid with a rheological relationship with fractional order derivatives of the Voigt and anti-Zener type with memory along the trajectory of the motion of a fluid. Theorems on the existence of the existences and uniqueness of a solution for degenerate parabolic and elliptical equations of high order describing  viscoelasticity models.

• Our researchers have produced theorems on the existence of solutions of the problem of  onlinear waves of established type on the surface of the ocean covered with ice.

• Our researchers have obtained an analog of the Liouville's theorem for axisymmetric flows of viscous fluids (stationary Navier-Stokes equations) with rotation.

• A description of the general structure of trajectory and global attractors of evolution equations with memory. In the proposed approach the dynamic system operates in  the space of the initial data of the Cauchy problem of the studied system. As an important application of the proposed method we have built a trajectory and a global attractor for a dissipative wave equation with linear memory. We have also proven the existence of a global Lyapunov function for a dissipative wave equation with memory. The existence of such a Lyapunov function allowed to prove the regularity of the structure of attractors that match nonstationary sets coming out of the set of stationary points if the reviewed equation.

• A proof of the theorem on the existence of a minimal trajectory pullback attractor and a global pullback attractor for both weak solutions of a nonautonomous medium with memory and for weak solutions of the model of the motion of a Bingham medium in the nonautonomous case.

• A proof of the theorems of the existence of weak solutions for the alpha-Leray model and the alpha-Navier-Stokes model with the viscosity coefficient depending on the temperature.

• Results concerning the interaction of the normal and the tangential operator into which the quadratic operator from the Helmholtz system decomposes. A theory of the nonlocal  stabilization of a system of the hydrodynamic type. A structure of stabilizing impulse control for the differentiated Burgers' equation, transfer of the built structure to the Helmholtz system.

• A theorem on the spectrum of the emerging operator pencil for a system of integro-differential equations describing the motion of a viscoelastic body fixed on the boundary of a bounded domain. A theorem on the multiple basicity of a system of root elements, a decomposition of the solution of the evolution problem. A theorem on the asymptotic of solutions.

• A review of the adiabatic mode for tilting ratchets. Implicit formulas have been obtained for the speed of the adiabatic migration of animals, fish or bacteria and the directions of their migration. It has been demonstrated that a longer range of the monotonicity of the potential defines the specific direction of migration. The result is based on a new nontrivial functional inequality. We have also studied the semiadiabatic mode for tilting ratchets, when the aggregate period of inclination tends to infinity and one of the states of inclination prevails over the other. We have obtained an explicit formula for the rate of semiadiabatic transfer and proven that if the potentials are nonconstant, semiadiabatic migration of animals, fish and bacteria occurs in the same direction. These results are in part based on one more functional inequality. For Brownian ratchets of arbitrary type with weak diffusion we have demonstrated that there is a connection between the transfer and some ordinary differential equation. If this ordinary differential equation does not have periodic solutions or, in other words, if its rotation number is not zero, an unidirectional flow of migration of animals, fish or bacteria emerges there, even though the average force is not zero.

• We have conducted field measurements of internal waves in August-September 2019 on the marine experimental station «Cape Schultz» of the V. I. Il'ichev Pacific Oceanological Institute of the Far Eastern Branch of the Russian Academy of Sciences, experimental data was collected. We have conducted a numerical modeling of nonlinear internal waves  and built a model of nonlinear internal waves (including Poincaré and Kelvin waves). A comparative analysis of the obtained experimental data and the results of a numerical modeling has been conducted.

• A research of the structure of the solutions and the solvability of the boundary value problem describing nonlinear waves in flows of continuously stratified fluid over an obstacle. On the basis of the model of weakly coupled Korteweg–de Vries equations  we have investigated the interaction of entangled traveling waves. We have built solutions of a number of reverse problems of the reproduction of the structure of nonlinear packages of internal waves. We have reconstructed the temperature fields and the boundaries of layers during the passage of a near-surface solitary wave and a tidal bore. We have conducted the verification of the built model by comparing it with experimental data collected not only at the marine experimental station «Cape Schultz» of the V. I. Il'ichev Pacific Oceanological Institute of the Far Eastern Branch of the Russian Academy of Sciences but also with experimental data collected in the South China Sea (Lien, Henyey, Ma and Yang, 2004).

• A definition of the theorem on the existence and uniqueness of a strong solution of the initial-boundary value problem for a system of equations of the motion of a viscoelastic fluid  that is a fractional counterpart of the Voigt viscoelasticity model  with memory along the spatial variable in the planar case. For a model of a thermo-viscoelastic medium of the Oldroyd type with memory along the trajectories of the motion we have determined weak solubility. In this case for the existence of trajectories of motion ensuring the memory of the medium we used the theory of regular Lagrange flows. A new method has been developed on the basis of the properties degenerate pseudo-differential operators built on a special integral transformation. On the basis of this method we researched new classes of pseudo-differential operators that allow to research the correctness of mathematical models of thermo-viscoelasticity  containing initial-boundary value and boundary value problwma for degenerate elliptical and  parabolic equations.

• A proof of the theorems on the weak solubility for models of viscoelastic media with  of a fractional order that is higher than 1.

• A proof of the convergence of the trajectory attractors and global attractors of approximations of the model of the motion of a viscoelastic medium with memory to the trajectory and global attractors of this model in the sense of semideviation while the approximation parameter tends to zero. Obtaining the convergence of trajectory and global attractors of approximations of the Bingham model to the trajectory and global attractors of the Bingham model in the sense of semiderivations while the approximation parameter tends to zero. A proof of the convergence of the trajectory and global attractors of the approximations of a modified Kelvin-Voigt model to the trajectory and global attractors of this model in the sense of semiderivations          while the approximation parameter tends to zero .

• A proof of the theorem on the solvability in the weak sense of the problem  of optimal control with feedback for Voigt models with variable density.

• Obtaining a theorem on the existence of global-in-time weak solutions of the initial-boundary value problem for the first-class alpha model of a Bingham model.

• A theorem on the existence of a weak solution of the problem of optimal control  with feedback for a modified Kelvin-Voigt model of weakly concentrated  water solutions of polymers.

• We have studied problems of rotationally symmetric flows in a bounded volume of a gas for all the values of the adiabat greater or equal than one. It has been proven that the matrix of concentrations is centered on the rotation axis. Moreover, it has been established that it has a unique non-zero element that lies on the diagonal and corresponds to the direction along the axis of symmetry OZ. The measure corresponding to this non-zero element is the product of a constant and the measure of length dz. In other words, the matrix of concentrations has only one non-zero element that is a Dirac measure concentrated on the axis of symmetry. It has also been demonstrated that the concentrations are absent in the case of axisymmetric solutions.

• A theorem on the global solvability of the initial-boundary value problem for multi-temperature multi-speed model of mixtures in the general case of three spatial variables.

• A proof, for the planar case, of the theorem on the existence and uniqueness of a strong solution of the initial-boundary value problem for systems of equations that describes the dynamics of a nonlinear thermo-viscoelastic continuous medium with a rheological relationship of the Oldroyd type.

• A proof of the theorem on the existence of strong solution of the problem of optimal control with feedback for a Navier-Stokes system with variable density in the planar case.

• Determining the theorem on the solvability, in the strong sense, of the problem of optimal control with feedback for a Voigt model with variable density.

• A proof of the theorem on the convergence of the sequence of solutions of a family of alpha-Bingham models to the solution of the original initial-boundary value problem when the parameter alpha tends to zero..

• Obtaining a formula for the monotonicity for the energy tensor for the problem of rotational symmetric flows of a viscous gas with the limiting value of the adiabatic indicator that equals to 1. It has been proven that outside of the axis of rotation the function of density allows for an estimate in the negative Sobolev space with an indicator of -1/2 that depends only on the data of the problem. We have proven the  existence of a weak solution defined outside of the rotation axis. It has been demonstrated that the concentration of the kinetic energy tensor is a symmetric and non-negative matrix Borel measure concentrated on the axis of rotation. It has been demonstrated that the matrix of the concentrations can have only one nontrivial component. We have proven that the time-averaging of the matrices of concentrations is completely continuous on the axis of rotation and its density is semicontinuous on the top.

• Obtaining a theorem on the weak solvability of the initial-boundary value problem for a heat-conducting multi-speed multi-dimensional model of the dynamics of mixtures accounting for the viscosity and comprehensibility.

• A theorem on the existence of weak solutions of initial-boundary value problems for equations of the motion for a model of a viscoelastic fluid with memory along the trajectory of the velocity field and a rheological relationship containing integer derivatives  of high orders.

• A proof of the weak solvability of initial-boundary value problems describing the motion of viscoelastic media (of the Voigt type) with lag coefficients dependent on temperature.

• A theorem on the existence of a weak solution of the initial-boundary value problem for a modified Kelvin-Voigt model with variable density.

• A statement on the existence of weak rotation-symmetric solutions of the magnetic gas dynamics accounting for the effects of viscosity and self-gravitation. A statement on the structure of possible concentrations of the energy tensor. An effective upper estimate for the critical value of the adiabatic index above which the concentrations are absent.

• A proof of the solvability of the regularization of the initial-boundary value problem for  equations of the dynamics of viscous of compressible multi-component  media.

Education and career development:

• In 2017 five employees of the Laboratory completed the additional training  program «Partial differential equations and their applications to mathematical hydrodynamics» at the Mathematics Department of the Middle East Technical University (Northern Cyprus).

• In 2018 five employees of the Laboratory completed additional training at the International Center for Mathematics (Portugal).

• In 2019 five employees of the Laboratory completed additional training with the topic «Multi-phase non-Newtonian fluids: mathematical modeling and computations» at a laboratory of the acadenuc institute of heat industry systems (France).

• We have staged international scientific conferences: «Modern methods and problems of mathematical hydrodynamics» (2017), «Modern methods and problems of mathematical hydrodynamics-2018» (2018), «Modern methods and problems   of mathematical hydrodynamics-2019» (2019).
• In 2017 we conducted the scientific seminar «Mathematical models of shallow-water shear flows» with an invited lecturer from Aix-Marseille University.
• Two Candidate of Science dissertations, two Doctor of Science dissertations, 6 master's degree theses and 5 bachelor's          degree theses  have been prepared and defended.
• The following lecture courses have been developed and introduced into the curriculum  of the Faculty of Mathematics of the Voronezh State University: «Applications of the theory of differential equations to geometry», «Navier-Stokes equations of compressible fluid», «Applications of differential inclusions to problems of optimal control», «Pavlovskiy mathematical models for the motion of polymer solutions», «Alpha models of equations of hydrodynamics», «Approximation-topological method for the solvability of equations of the dynamics of viscoelastic media».
• Two study guides have been published. 

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..