### Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions : slow oscillations of beams on elastic bearings

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In this note, we discuss several recently developed methods for studying stability of a singular limit process with respect to the shape of the underlying physical space. As a model example, we consider a compressible viscous barotropic fluid occupying a spatial domain $\Omega \subset {R}^{3}$. In what follows, we describe two rather different problems: (i) the choice of effective boundary conditions; (ii) the fluid flow in the low Mach number regime. In the remaining part of the paper, we analyze these two issues simultaneously...

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.

In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation ${U}_{tt}+d{U}_{t}-\sigma {(x,t,{U}_{x})}_{x}+aU=f(x,t,{U}_{x},{U}_{t},U)$ with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function $f$. The main idea of the proof relies on the compensated compactness theory.

The author investigates time-periodic solutions of the quasilinear beam equation with the help of accelerated convergence methods. Using the Newton iteration scheme, the problem is approximated by a sequence of linear equations solved via the Galerkin method. The derivatiove loss inherent to this kind of problems is compensated by taking advantage of smoothing operators.

The existence of small global (in time) solutions to an abstract evolution equation containing a damping term is proved. The result is then applied to fully nonlinear telegraph equations and to nonlinear equations involving operators with time delay.

The author examined non-zero $T$-periodic (in time) solutions for a semilinear beam equation under the condition that the period $T$ is an irrational multiple of the length. It is shown that for a.e. $T\in {R}^{1}$ (in the sense of the Lebesgue measure on ${R}^{1}$) the solutions do exist provided the right-hand side of the equation is sublinear.

In the paper, time-periodic solutions to dynamic von Kármán equations are investigated. Assuming that there is a damping term in the equations we are able to show the existence of at least one solution to the problem. The Faedo-Galerkin method is used together with some basic ideas concerning monotone operators on Orlicz spaces.

In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.

In the paper, we deal with the equation of a rectangular thin plate with a simply supported boundary. The restoring force being an odd superlinear function of the vertical displacement, the existence of infinitely many nonzero time-periodic solutions is proved.

A parabolic system arisng as a viscosity regularization of the quasilinear one-dimensional telegraph equation is considered. The existence of $L\infty $ - a priori estimates, independent of viscosity, is shown. The results are achieved by means of generalized invariant regions.

We show compactness of bounded sets of weak solutions to the isentropic compressible Navier-Stokes equations in three space dimensions under the hypothesis that the adiabatic constant $\gamma >3/2$.

This is a survey of some recent results on the existence of globally defined weak solutions to the Navier-Stokes equations of a viscous compressible fluid with a general barotropic pressure-density relation.

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