×

Convex dynamics in Hele-Shaw cells. (English) Zbl 1023.30018

In this interesting paper at first the authors give an overview about the development of the theory of the one-phase Hele-Shaw problem in two dimensions. Then they study the convex and starlike dynamics of the free boundary geometry for the injection problems with a constant source at infinity (contracting bubble) and in a finite point, respectively. For the contracting bubble they prove that for a starlike initial domain with analytic boundary the starlikeness is preserved locally in time in case of small surface tension, and in case of no surface tension so long as the solution exists where the rate of the area change is controlled by variation of the square of bubble capacity. For the second problem where the convex dynamics is preserved locally in time if the initial domain is \(\alpha\)-convex, the authors derive some corresponding isoperimetric inequalities.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
76D99 Incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI EuDML