Existence and uniqueness of classical solutions for the multidimensional expanding Hele–Shaw problem are proved. 1. The problem. We are concerned with a class of moving boundary problems for bounded domains in R n , which comprise in particular the so-called single phase Hele–Shaw problem. In order to describe precisely the involved geometry, let Ω be a bounded domain in R n and assume that its...

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker–Planck equations ρt−Δβ(ρ)+div(Db(ρ)ρ)=0 in (0,∞)×Rd, ρ(0,x)≡ρ0(x). Under suitable assumptions on β:R→R,b:R→R D:Rd→Rd, d≥1, this equation generates a unique flow ρ(t)=S(t)ρ0:[0,∞)→L1(Rd) as mild solution sense nonlinear semigroup theory. also class L∞((0,T)×R...

For a set X , an equivalence relation ρ on X , and a cross-section R of the partition X/ρ induced by ρ, consider the semigroup T (X, ρ, R) consisting of all mappings a from X to X such that a preserves both ρ (if (x, y) ∈ ρ then (xa, ya) ∈ ρ) and R (if r ∈ R then ra ∈ R). The semigroup T (X, ρ, R) is the centralizer of the idempotent transformation with kernel ρ and imageR. We determine the str...