Elena Vuk

On the extensible viscoelastic beam

This work is focused on the equation describing the motion of an extensible viscoelastic beam. Under suitable boundary conditions, the related dynamical system in the history space framework is shown to possess a global attractor of optimal regularity. The result is obtained by exploiting an appropriate decomposition of the solution semigroup, together with the existence of a Lyapunov functional.

Long-Term Damped Dynamics of the Extensible Suspension Bridge

This work is focused on the doubly nonlinear equation 𝜕 𝑡 𝑡 𝑢 + 𝜕 𝑥 𝑥 𝑥 𝑥 𝑢 + ( 𝑝 − ‖ 𝜕 𝑥 𝑢 ‖ 2 𝐿 2 ( 0 , 1 ) ) 𝜕 𝑥 𝑥 𝑢 + 𝜕 𝑡 𝑢 + 𝑘 2 𝑢 + = 𝑓 , whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness 𝑘 2 . When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load 𝑝 and stiffness 𝑘 2 . For a general external source 𝑓 , we prove the existence of bounded absorbing sets. When 𝑓 is time-independent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem.

Long-term dynamics of the coupled suspension bridge system

In this paper we study the long-term dynamics of a nonlinear suspension bridge system. The road bed and the main cable are modeled as a nonlinear beam and a vibrating string, respectively, and their coupling is carried out by one-sided springs. First, we scrutinize the set of stationary solutions, which turns out to be nontrivial when the axial load exceeds some critical value. Then, we prove the existence of a bounded global attractor of optimal regularity and we give its characterization in terms of the steady states of the problem.

Steady-state solutions for a suspension bridge with intermediate supports

This work is focused on a system of boundary value problems whose solutions represent the equilibria of a bridge suspended by continuously distributed cables and supported by M intermediate piers. The road bed is modeled as the junction of N=M+1 extensible elastic beams which are clamped to each other and pinned at their ends to each pier. The suspending cables are modeled as one-sided springs with stiffness k. Stationary solutions of these doubly nonlinear problems are explicitly and analytically derived for arbitrary k and a general axial load p applied at the ends of the bridge. In particular, we scrutinize the occurrence of buckled solutions in connection with the length of each sub-span of the bridge.MSC:35G30, 74G05, 74G60, 74K10.

Uniform attractors for a semilinear evolution problem in hereditary simple fluids

Abstract This paper is devoted to study the asymptotic behavior, as time tends to infinity, of the solutions of a semilinear partial differential equation of hyperbolic type with a convolution term describing simple fluids with fading memory. The past history of the displacement is regarded itself as a new variable, so that the corresponding initial boundary value problem is transformed into a dynamical system in a history space setting. Under proper assumptions on the nonlinear term, the existence of an uniform absorbing set and an universal attractor in suitable function spaces are achieved.

Extremum principles in electromagnetic systems

Variational expressions and saddle-point (or “mini-max”) principles for linear problems in electromagnetism are proposed. When conservative conditions are considered, well-known variational expressions for the resonant frequencies of a cavity and the propagation constant of a waveguide are revised directly in terms of electric and magnetic field vectors. In both cases the unknown constants are typefied as stationary (but not extremum) points of some energy-like functionals. On the contrary, if dissipation is involved then variational expressions achieve the extremum property. Indeed, we point out that a saddle-point characterizes the unique solution of Maxwell equations subject to impedancelike dissipative boundary conditions. In particular, we deal with the quasi-static problem and the time-harmonic case.