The existence of a smooth interface in the evolutionary elliptic Muskat--Verigin problem with nonlinear source

TL;DR

This paper proves the local-in-time existence of smooth solutions and free boundaries for a two-phase elliptic Muskat--Verigin problem with nonlinear sources, using parabolic regularization techniques.

Contribution

It introduces a novel approach to establish local existence of smooth solutions and free boundaries in a nonlinear elliptic two-phase problem.

Findings

Existence of smooth solutions and free boundaries is established.

Method employs parabolic regularization of free boundary conditions.

Results are valid locally in time.

Abstract

We study the two-phase Muskat--Verigin free-boundary problem for elliptic equations with nonlinear sources. The existence of a smooth solution and a smooth free boundary is proved locally in time by applying the parabolic regularization of a condition on the free boundary.