An integral domain $R$ is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element $a$ in $R$ , the ascending chain of non-associate irreducible divisors in $R$ of ${{a}^{n}}$ stabilizes on a finite set as $n$ ranges over the positive integers, while $R$ is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension $S$ of $R$ is a root extension or radical extension if for each $s$ in $S$ , there exists a natural number $n\left( s \right)$ with ${{s}^{n\left( s \right)}}$ in $R$ . In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $\left( R,\,S \right)$ is governed by the relative sizes of the unit groups $\text{U}\left( R \right)$ and $\text{U}\left( S \right)$ and whether $S$ is a root extension of $R$ . The following results are deduced from these considerations: An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let $R$ be a Noetherian domain with integral closure $S$ . Suppose the conductor of $S$ into $R$ is non-zero. Then $R$ is IDPF if and only if $S$ is a root extension of $R$ and $\text{U}\left( S \right)/\text{U}\left( R \right)$ is finite.

Flows with free boundaries in a Hele-Shaw cell provide a unique opportunity to study non-linear boundary dynamics using rigorous analytic approaches. While of limited direct ‘practical value’, these studies give rise to a plethora of new phenomena and insights which may serve as beacons in the turbulent ocean of moving free boundaries and pattern forming. This paper gives a brief summary of the authors' studies of Hele-Shaw flows with free boundaries and some related problems based upon Richardson's approach. Some promising directions of further research are also discussed.

Generalized two-phase fluid flows in a Hele-Shaw cell are considered. It is assumed that the flow is driven by the fluid pressure gradient and an external potential field, for example, an electric field. Both the pressure field and the external field may have singularities in the flow domain. Therefore, combined action of these two fields brings into existence some new features, such as non-trivial equilibrium shapes of boundaries between the two fluids, which can be studied analytically. Some examples are presented. It is argued, that the approach and results may find some applications in the theory of fluids flow through porous media and microfluidic devices controlled by electric field.