Convex dynamics in Hele-Shaw cells

Abstract

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.