In this paper we discuss the existence and asymptotic behavior of Kneser solutions of nth-order two term quasilinear ordinary differential equations which are generalizations of the Emden-Fowler equation. It will be shown that there is a simple and significant difference between the super-homogeneous case and the sub-homogeneous case for the existence and asymptotic behavior of Kneser solutions of these equations.

We derive two generalizations of Gasper's transformation formula for basic hypergeometric series. Using these generalized transformation formulas, we give explicit expressions for the coefficients of three-term relations for the basic hypergeometric series ₂φ₁, which are generalizations of the author's previous results on three-term relations for ₂φ₁. These generalized three-term relations involve multiplying the variable of one of the three ₂φ₁ by a power of q. Also, as corollaries of the generalizations of Gasper's transformation formula, we derive a few generalizations of Gasper's summation formulas for basic hypergeometric series.

We consider a two-dimensional viscous incompressible fluid in interaction with a one-dimensional beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result, where the initial deformation of the beam was assumed to be small. Here our result is valid for any smooth initial deformation (W^7,∞) with no restriction on the size. The main point of the proof consists in the study of the linearized system in a non-cartesian domain and in particular in proving that the corresponding semigroup is of Gevrey class.

We consider the large time behavior of the three-dimensional Navier-Stokes flow around a rotating rigid body. Assume that the angular velocity of the body gradually increases until it reaches a small terminal one at a certain finite time and it is fixed afterwards. We then show that the fluid motion converges to a steady solution as time t → ∞.