An Elliptic-Parabolic Free Boundary Problem with Discontinuous Data

TL;DR

This paper studies a 1+1 dimensional elliptic-parabolic free boundary problem modeling fluid flow in porous media, showing how boundary and initial data discontinuities affect solution and free boundary regularity, with optimal regularity results.

Contribution

It provides the first detailed regularity analysis of solutions and free boundaries under discontinuous data for this class of elliptic-parabolic free boundary problems.

Findings

Weak solutions are Lipschitz in space and $C^{1/2}$ in time.

The free boundary is locally a $C^{1/2}$ graph.

Regularity results are optimal.

Abstract

We consider an elliptic-parabolic free boundary problem that models the fluid flow through a partially saturated porous medium. The free boundary arises as the interface separating the saturated and unsaturated regions. Our main goal is to investigate, for the 1+1 dimensional model, how jump discontinuities on the boundary and initial data influence the regularity of both the solution and the free boundary. We show that if the data is merely bounded, then weak solutions are Lipschitz in space and $C^{1/2}$ in time in the unsaturated region. Moreover, the free boundary is locally the graph of a $C^{1/2}$ function, and this regularity is optimal. We view this analysis as a stepping stone towards the study of local regularization for higher-dimensional elliptic-parabolic free boundaries.