Aloisio Neves

On the spectrum of evolution operators generated by hyperbolic systems

Abstract We prove that the asymptotic behavior of semigroups generated by mixed-initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation. The proof is based on the following reduction principle: we construct a reduced system (which is much simpler than the given one) such that the difference of the semigroups is a compact operator. The construction of the reduced system is presented in the non-autonomous case because it has implifications in the study of systems which are time periodic.

Periodic motions of sucker rod pumping systems

AMONG the most important of the artificial lifting methods used in petroleum industries are the sucker rod pumping systems. In order to improve the design and the operation of such installations, it is necessary to understand their behavior, especially at infield situations. The pumping system is put into action by an engine that imparts a periodic motion to the top of the rods, and eventually all of the system sets itself in a periodic motion. To describe this behavior, several mathematical models have been developed over the years by taking into consideration various inertial and dissipation effects in the system, material properties, down hole pumping conditions, fluid interactions, and so on (see for instance Doty and Schmidt [l]). Obviously, several nonlinearities are present in the problem, but there is one that is intrinsic to the pumping system itself. This is due to the presence of the valve mechanism that imposes sudden changes in the stress acting on the rods according to the opening or the closing of the down hole and walking valves. Concerning the study of these models, some theoretical analysis has been done in certain linearized versions of the models, and many numerical experiments were performed. In general, however, there is a lack of sound mathematical analysis of the nonlinear models. In this paper we consider a simple model that takes into consideration the elastic response of the rods, the dissipation mechanism (due to internal material dissipation and hydraulic friction) and the stress change due to the valve mechanism. It corresponds to the same partial differential equation used in Pavlik and Schafer [2], it is a wave equation with dissipation, but with a nonlinear and discontinuous boundary condition modeling the action of the valves. Our objective is to study the interaction between the action of the valves with the periodic motion imparted by the engine on the top of the rods in the production of the final motion of the rod. We are able to show the existence of a strong forced periodic solution for the mathematical model of this problem in the following way: we first show that the operator associated to the problem is maximal monotone in a convenient Hilbert space. Thus, we are in the setting of the well-known theory of monotone operators, but we cannot apply its standard results on periodic solutions because our operator is neither strictly monotone nor coercive. The main difficulty is to control the influence of the boundary conditions. We then proceed by approximating our equation in a suitable way by equations for which the corresponding operators are strictly coercive; for them the standard theory can be applied, and approximated periodic solutions are obtained. We then strive to obtain certain estimates for these approximated solutions, which permit us to conclude that they approach a mild periodic solution of the original problem. An essential step for this is the construction of an adequate Liapunov function that furnishes an