On the Frequency Drift of Coronal Loop’s Fast Kink Oscillation: Effects of Quasi-static Evolution in Loop Density

Although the fast kink oscillation, as one of a few fundamental modes in coronal seismology, has received a lot of attention over the past two decades, observations of its frequency drift remain elusive. There is evidence that this phenomenon is related to the quasi-static evolution of loop density. We therefore consider analytically the effects of a quasi-static density evolution on the fast kink oscillation of coronal loops. From the analyses, we determine explicitly the analytic dependence of the oscillation period/frequency and amplitude on the evolving density of the oscillatory loop. The findings can well reconcile several key characters in some frequency drift observations, which are not understood. Models of fast kink oscillation in the thermal dynamic loop are also established to investigate the present effects in more detail. Our findings not only show us a possible explanation for the frequency drift of the coronal loop’s fast kink oscillation, but also a full new energy transformation mechanism where the internal energy and the kinetic energy of an oscillating coronal loop can be interchanged directly by the interaction of the loop’s oscillation and its density evolution, which we suggest may provide a new clue for the energy processes associated with a thermodynamic resonator in the space magnetic plasma.

Analyticity of the Resolvent Function

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

Existence results for a nonlinear nonautonomous transmission problem via domain perturbation

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$ . First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$ .