Large-time rescaling behaviors of some rational type solutions to the Polubarinova-Galin equation with injection

الأصل · EN

The main goal of this paper is to give a precise description of rescaling behaviors of rational type global strong solutions to the Polubarinova-Galin equation. The Polubarinova-Galin equation is the reformulation of the zero surface tension Hele-Shaw problem with a single source at the origin by considering the moving domain as the Riemann mapping of the unit disk centered at the origin. The coefficients {aₖ(t)}ₖ≥ ₂ of the polynomial strong solution fₖ₀(ξ,t)=∑ᵢ₌₁ᵏ₀aᵢ(t)ξⁱ decay to zero algebraically as t⁻λₖ (λₖ=k/2) and the decay is even faster if the low Richardson moments vanish. The dynamics for global solutions are discussed as well.