2016/05/24 by Héctor A. Chang‐Lara, Héctor A. Chang-Lara, Chang-Lara, Héctor A. +2
Computer Science · Mathematics · #35B65 #35R09 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35B65 #msc:35R09 #msc:35R35
paper · pdf · doi:10.48550/arxiv.1605.07591
48 pages, 5 figures
arxiv created 2016/05/24 · openalex publication_date 2016/05/24 · arxiv updated 2016/05/25 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary --represented by the hodograph transform of the solution- solves a nonlinear integro-differential equation. This nonlinear equation is linearized to a (nonlocal) parabolic equation with bounded measurable coefficients, for which regularity estimates are available. This fact is used to prove a regularity result for the free boundary of a weak solution near points where the solution looks sufficiently flat. More concretely, flat means that in a parabolic neighborhood of the point the solution lies between the solutions corresponding to two parallel flat fronts a small distance apart --a condition that only depends on the the local behavior of the solution. In a neighborhood of such a point, the free boundary is given by the graph of a function whose spatial gradient enjoys a universal Hölder estimate in both space and time.