Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

Abstract.

We consider multidimensional weak and strong Hele-Shaw dynamics Ω(t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain Ω(0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions Ω(t), and show that if Ω(0) is starshaped with respect to a small ball centered on the point of injection, then the evolution Ω(t) exists for all time.