Victor G. Zvyagin

Optimal feedback control in the problem of the motion of a viscoelastic fluid

We study an optimization problem for the feedback control system emerging as a regularized model for the motion of a viscoelastic fluid subject to the Jeffris-Oldroyd rheological relation. The approach includes systems governed by the classical Navier-Stokes equation as a particular case. Using the topological degree theory for condensing multimaps we prove the solvability of the approximating problem and demonstrate the convergence of approximate solutions to a solution of a regularized one. At last we show the existence of a solution minimizing a given convex, lower semicontinuous functional.

On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications

We define a nonoriented coincidence index for a compact, fundamentally restrictible, and condensing multivalued perturbations of a map which is nonlinear Fredholm of nonnegative index on the set of coincidence points. As an application, we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.

An oriented coincidence index for nonlinear Fredholm inclusions with nonconvex-valued perturbations

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion, is given.

Optimal Feedback Control in the Mathematical Model of Low Concentrated Aqueous Polymer Solutions

We consider the feedback control problem in the model of motion of low concentrated aqueous polymer solutions. We demonstrate the solvability of an approximating problem, using some a priori estimates and the topological degree theory. Then the convergence (in some generalized sense) of solutions of approximating problems to a solution of the given problem is proved. Moreover, we show the existence of a solution minimizing a given convex, lower semicontinuous functional.

On the weak solvability of one system of thermoviscoelasticity

The weak solvability of the initial-boundary value problem for one type model of motion of a thermoviscoelastic continuum is studied. The proof of the main result is based on the construction of an iterative process which consists of the successive solution of the dynamical part of system of the motion equations and the heat conduction equation. The solvability of the first system is obtained by means of the approximation-topological method, based on a priori estimates and the Leray-Schauder degree theory. For the solvability of the heat conduction equation which contains L1 components we use the theory of analytic semigroup and fractional powers of operators.

On weak solutions for generalized Oldroyd model for laminar and turbulent flows of nonlinear viscous-elastic fluid

We consider the statement of an initial-boundary value problem for a generalized Oldroyd model describing both laminar and turbulent motions of a nonlinear viscous-elastic fluid. The operator interpretation of a posed problem is presented. The properties of operators forming the corresponding equations are investigated. We introduce approximating equations and prove their solvability. On that base the existence theorem for the operator equation equivalent to the stated initial-boundary value problem is proved.

Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle

The qualitative dynamics of weak solutions to a nonautonomous model of polymer solution motion (with a rheological relation satisfying the objectivity principle) is studied using the theory of pullback attractors of trajectory spaces. For this purpose, the existence of weak solutions is proved for the model under study, a family of trajectory spaces is defined, trajectory and minimal pullback attractors are introduced, and their existence is proved.

Topological approach to investigation of acoustic axes in crystals

The concept of topological degree of a map is generalized to the case of discontinuous maps. The numerical value of such a degree may be a rational number. The representations developed are used for topological interpretation of the characteristics of special directions of propagation of acoustic waves in crystals (specifically, acoustic axes). The Euler theorem is generalized to the case of singularities with rational indices, and this result is applied to the set of acoustic axes in crystals.

On problem of the dynamics of a viscoelastic medium with memory on an infinite interval

The existence of a weak solution of a boundary value problem for a viscoelasticity model with memory on an infinite time interval is proved. The proof relies on an approximation of the original boundary value problem by regularized ones on finite time intervals and makes use of recent results concerning the solvability of Cauchy problems for systems of ordinary differential equations in the class of regular Lagrangian flows.

Weak solvability of a fractional Voigt viscoelasticity model

The existence of a weak solution of a boundary value problem for a fractional Voigt viscoelasticity model is proved. The proof relies on an approximation of the original boundary value problem by regularized ones and recent results concerning the solvability of Cauchy problems for systems of ordinary differential equations in the class of regular Lagrangian flows.