Infinite lifetime for the starlike dynamics in Hele-Shaw cells

Source abstract

One of the “folklore" questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain Ω 0 \Omega _0 , the Hele-Shaw chain of subordinating domains Ω ( t ) \Omega (t) , Ω 0 = Ω ( 0 ) \Omega _0=\Omega (0) , exists for an infinite time under injection at the point of starlikeness.