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Registration Number

0214U007984, 0112U001062 , R & D reports

Title

Qualitative behavior of solutions to dissipative evolution coupled partial differential equations

Head

Chueshov Igor,

Registration Date

19-01-2015

Organization

Kharkov National University named after V.N. Karazin

Description

Objects of the research are initial-boundary value problems for the following systems: coupled nonlinear stochastic equations; models describing oscillations of coupled systems "nonlinear plate + fluid" (for the cases of transversal oscillations of the plate only and for simultaneous transversal and in-plane oscillations of the plate); quasilinear Kirchhoff equation with nonlinear damping; Schr?dinger-Boussinesq system. A goal of the work is to study synchronization phenomena for coupled nonlinear stochastic equations using inertial manifolds; to prove well-posedness of systems, describing coupled oscillations of plate and fluid and to study asymptotical behaviour of these systems; to investigate well-posedness and asymptotical behaviour, to study attractor structure for quasilinear Kirchhoff equation and Schr?dinger-Boussinesq system. The research is performed in framework of dynamical systems theory. At the first stage well-posedness of initial-boundary value problem for quasilinear Kirchhoff equation with nonlinear dissipation was proved for initial data from appropriate functional space, a-priory bounds where obtained and energy equality was proved. Also for this model existence of compact global attractor was established (in "stronger" phase space in the supercritical case). In subcritical case the result was improved: it was proved that trajectories tend to attractor with respect to energy space topology. Moreover, in this case existence of fractal exponential attractor was proved and conditions of existence of finite set of determining functionals were obtained. At the second stage dynamics of several coupled systems "plate + fluid" was investigated. The first system consists of 3D Navier-Stocks equations linearized near Poiseulle flow in an unbounded domain and classical (possibly nonlinear) elastic plate equation for transversal displacement at the elastic part of the boundary. It was proved that this problem generates evolution semigroup at the corresponding phase space and compact finite-dimensional global attractor exists for the semigroup. It was also proved that the semigroup is exponentially stable С0 semigroup of linear operators in the purely linear case. Since no mechanical damping was accounted for in the system, this means energy dissipation due to viscosity in the fluid flow is sufficient for stabilization of the entire system. Similar results were obtained for the coupled system in bounded domain, that consists of linearized Navier-Stocks equations and classical nonlinear elastic plate equation for transversal displacement at the elastic flat part of the boundary. In this case the results have place under weaker assumptions about the system. Also we considered the problem of well-posedness and asymptotical behaviour of the coupled system consisting of linearized 3D Navier-Stocks equations and classical nonlinear full von Karman equation for shallow shell, that describes both transversal and in-plane displacement on the elastic part of the boundary. Rotational inertia of the filaments of the shell was accounted for. It was proved, that the equation system has global solution and generates a dynamical system in the corresponding phase space. Also existence of a compact global attractor for the dynamical system was proved for the case of dissipative term included in the transversal displacement equation and under certain structure of external loads. At the third stage Schr?dinger-Boussinesq system and coupled nonlinear stochastic system were considered. Well-posedness of Schr?dinger-Boussinesq system was proved in the corresponding phase spaces and conditions on nonlinear term under which a compact global attractor exists were established. In the case of certain structure of external loads the reduction principle (to the Boussinesq equation) was proved. For the coupled nonlinear stochastic system conditions on equation operators and stochastic terms under which the system is well-posed and asymptotical synchronization has place were established. The corresponding random manifold which describes synchronization character was constructed. Also additional requirements were established under which the system synchronizes exponentially quickly. The methods of the research are theoretical considerations. The research is of theoretical nature. Its results can be applied for prediciton of long time behaviour of various vibration machines. The methods developed in the research can be used for investigation of the other systems of coupled equations.

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Болтоносов О.І.
Колбасін С.О.
Рибалко В.О.
Рижкова-Герасимова І.А.
Чуєшов І.Д.
Щербина М.В.
Щербина О.С.