Hyperbolic Limit Sets of Evolutionary Equations and the Galerkin Method

Abstract

The angular correlation of the two gamma quanta emitted when a thermalised positron annihilates with metallic conduction electrons is investigated by applying the newly developed theory in which an electron gas is considered as a system of interacting collective excitations. The conduction electrons are treated in a jellium model. The method leads in a natural way to the appearance of high-momentum components (i.e. pair momentum P > pF) in the annihilation radiation. The amount of these components is significant ( 2 lo",) in a dilute electron gas (such as in alkali metals), but fairly irrelevant for higher densities. The dependence on momentum of the enhancement factor for a dense system (wzith r, = 2 ) agrees well both with the earlier theories given b) Kahana and others, and also with recent accurate experimental observations. As r, increases into the alkali-metal region. the enhancement factor for P c. pF becomes relatively more and more constant, in contrast with the trend in the Kahana theory. In this density regime the experimental results vary widely. although most of them disagree with the present prediction. The additional effects due to core annihilation and lattice periodicity are studied and reviewed.

Perturbations of foliated bundles and evolutionary equations

“…First we extend the perturbation theory to include foliated invariant sets in an infinite-dimensional setting, including the Navier-Stokes equations. (This extension unifies the earlier work on compact invariant manifolds, see [24,5], on the one hand, and the work on hyperbolic sets, see [15], on the other.) Second, we treat both the autonomous and the nonautonomous problems in this new perturbation theory.…”

Hopf's conjecture for a class of chemical kinetics equations

“…Then, we shall formulate the fundamental properties of this system. The proofs of all the statements given below are standard and therefore we shall restrict ourselves only to sketch them and we refer the reader to [6][7][8][9] where similar questions are discussed in a more detailed manner.…”

“…In the space W ~ we consider the linear evolution equation &~ (2 .i) and can be carried out according to the scheme presented in [6] (see [2,9]). If ~e~k , then Z(~)= ~ and, consequently, on an invariant set, the dynamical system [%} ceases to be local relative to time.…”

Approximation of attractors of semidynamical systems

“…Starting from [9], [10] the problem of the convergence of M λ to M has been studied by many authors. Conditions ensuring that the attractor of the original problem depends upper semicontinuously on the perturbation parameter in the solution operator of the problem have been found.…”