Large-time rescaling behaviors for large data to the Hele-Shaw problem

This paper addresses rescaling behaviors of some classes of global solutions to the zero surface tension Hele-Shaw problem with injection at the origin, {Ω(t)}t≥0. Here Ω(0) is a small perturbation of f(B1(0), 0) if f(ξ, t) is a global strong polynomial solution to the Polubarinova-Galin equation with injection at the origin and we prove the solution Ω(t) is global as well. We rescale the domain Ω(t) so that the new domain Ω ′ (t) always has area π and we consider ∂Ω ′ (t) as the radial perturbation of the unit circle centered at the origin for t large enough. It is shown that the radial perturbation decays algebraically as t. This decay also implies that the curvature of ∂Ω ′ (t) decays to 1 algebraically as t. The decay is faster if the low Richardson moments vanish. We also explain this work as the generalization of Vondenhoff’s work which deals with the case that f(ξ, t) = a1(t)ξ.