International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 11, Pages 639-650

Convex dynamics in Hele-Shaw cells

Dmitri Prokhorov1 and Alexander Vasil'ev2

1Department of Mathematics and Mechanics, Saratov State University, Saratov 410026, Russia
2Departamento de Matemática, UTFSM, Casilla, Valparaíso 110-V, Chile

Received 15 March 2002

Copyright © 2002 Dmitri Prokhorov and Alexander Vasil'ev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study geometric properties of a contracting bubble driven by a homogeneous source at infinity and surface tension. The properties that are preserved during the time evolution are under consideration. In particular, we study convex dynamics of the bubble and prove that the rate of the area change is controlled by variation of the bubble logarithmic capacity. Next we consider injection through a single finite source and study some isoperimetric inequalities that correspond to the convex and α-convex dynamics.